Bloch wave framework for structures with nonlocal interactions: Application to the design of origami acoustic metamaterials

We present a generalized Bloch wave framework for the dynamic analysis of structures with nonlocal interactions and apply it to the design of origami acoustic metamaterials. Specifically, we first discretize the origami structures using a customized structural bar-and-hinge model that minimizes the degrees of freedom in the associated unit cell, while being sufficiently accurate to capture the behavior of interest. Next, observing that this discretization results in nonlocal structural interactions-the stiffness matrix has nonzeros between nodes that are not nearest neighbors due to the coupled deformations arising during folding or bending-we generalize the standard Bloch wave approach used in structural analysis to enable the study of such systems. Utilizing this framework, choosing the geometry of the unit cell as well as the folded state of the structure as design variables, we design tunable and programmable Miura-ori and eggbox strips, sheets, and composites that are large band, low frequency acoustic switches. In doing so, we find that the number of bandgaps in the sheets is significantly smaller than their strip counterparts and also occur at relatively higher frequencies, a limitation which is overcome by considering composite structures that have individual panels made of different materials. Overall, we have found origami structures to be ideal candidates as acoustic metamaterials for noise control, vibration isolation, impact absorption, and wave guides.