Lattice reconfiguration and phononic band-gap adaptation via origami folding

We introduce a framework of utilizing origami folding to redistribute the inclusions of a phononic structure to achieve significant phononic band-gap adaptation. Cylindrical inclusions are attached to the vertices of a Miura-Ori sheet, whose 1 degree-of-freedom rigid folding can enable fundamental reconfigurations in the underlying periodic architecture via switching between different Bravais lattice types. Such a reconfiguration can drastically change the wave propagation behavior in terms of band gap and provide a scalable and practical means for broadband wave tailoring.

Figure 1

(a–c) Illustration of folding-induced topology transformation in origami structures (OS). Circular cross-section solid cylindrical inclusions are attached, connecting the vertices of rigid-foldable Miura-Origami sheets with air as the host media; cross-sectional plots in bottom row clearly illustrate topology transformation during folding operation. (d) Miura-Ori unit-vertex parameters.

Figure 2

Achievable Bravais lattice types via Miura-Ori rigid folding based on different crease designs. (a) Combinations of γ and $a/b$ that can achieve center-rectangular (CR) lattice type. White regions correspond to Miura-Ori designs that cannot achieve CR, and the shaded regions correspond to crease designs that can achieve CR multiple times as the Miura folds from $\theta =0^{\circ }$ to 90°. For visual clarity, demarcation between Miura-Ori designs that can achieve CR more than 3 times during folding are not plotted separately. Dashed curves in this plot correspond to the subsets of γ and $a/b$ combinations where one of the achievable CR lattice types is actually hexagonal (H). Again for clarity, only the curves based on $n=1-6$ are plotted. (b) Similar plot as (a), but for rectangular (R) and square (S) lattice types. When considering the topology transformation of a specific Miura-Ori design, both of the two plots need to be considered. In (c–f), we show how folding shifts the lattice configuration between square (S) and Hexagon (H) in OS with a triangular crease design (marked by triangular marker in a and b). Effect of folding on topology transformations corresponding to star, circle, polygon, and square crease designs are provided in Appendix pp3.

Figure 3

3D printed origami sheet. Top view of (a) individual parts and (b) assembled origami sheet. (c–e) Instances of assembled unit cell of hexagonal-OS, illustrating transformation between different lattice types/topologies. (c) Hexagon lattice at 0°. (d) Square lattice at 55°. (e) Hexagon lattice at 70°. (f–h) Reflect the instances of Supplemental Movie [37] S1 at different folding angles corresponding to (c–e).

Figure 4

(a) Band structure of acoustic wave propagation in hexagonal-OS at 65° folding angle with three complete PBGs (horizontal shaded regions). 2D cross section along transverse plane with representative unit cell and first Brillouin zone are given in the inset. (b) Effect of folding on PBGs of hexagonal-OS. Square (circle) markers represent the lower (upper) edge frequencies of PBG. Lattice topology and unit cell (marked in dashed-shaded region) at 0°, 55°, and 70° folding angle are given in the inset (left to right), each corresponding to hexagon, square, and hexagon lattice types respectively.

Figure 5

Comparison of low-frequency hexagonal-OS PBGs with that of square and hexagonal lattice types. The lower (upper) edge frequencies of PBGs are represented by square (circle) markers.

Figure 6

PBG adaptation in OS with honeycomb as initial lattice configuration. Square (circle) markers represent the lower (upper) edge frequencies of PBG. Distribution of solid circles in the inset represent the lattice topology and dashed-shaded region represents the unit cell of honeycomb-OS at 0°, 55°, and 70° folding angles (left to right).

Figure 7

(a–e) Five types of 2D Bravais lattice types.

Figure 8

(a–b) Examples illustrating two hexagonal lattices where (a) $n=1$ and (b) $n=2$. Even though the hexagonal lattices are identical between the two examples, the underlying Miura-Ori designs have different geometric design parameters.

Figure 9

(a–e) Topology transformation in OS with different Miura-Ori crease designs marked in Figs. 2 and 2.

Figure 10

Wave propagation characteristics evaluated via PWE method (right solid line) and COMSOL (left dashed line) along ΓX wave vector direction. The band gaps evaluated via PWE method along $X$ direction represented by horizontal gray regions match very well with transmission dips generated via comsol. (A,B,C) Acoustic pressure plots at different instances along the transmission spectra demonstrating wave propagation in band gap and band pass transmission regions.