Formation of rarefaction waves in origami-based metamaterials

We investigate the nonlinear wave dynamics of origami-based metamaterials composed of Tachi-Miura polyhedron (TMP) unit cells. These cells exhibit strain softening behavior under compression, which can be tuned by modifying their geometrical configurations or initial folded conditions. We assemble these TMP cells into a cluster of origami-based metamaterials, and we theoretically model and numerically analyze their wave transmission mechanism under external impact. Numerical simulations show that origami-based metamaterials can provide a prototypical platform for the formation of nonlinear coherent structures in the form of rarefaction waves, which feature a tensile wavefront upon the application of compression to the system. We also demonstrate the existence of numerically exact traveling rarefaction waves in an effective lumped-mass model. Origami-based metamaterials can be highly useful for mitigating shock waves, potentially enabling a wide variety of engineering applications.

Figure 1

(a) Flat front and rear sheets of the TMP with mountain (solid lines) and valley (dashed lines) crease lines. (b) Folding motion of the TMP unit cell. Shaded area is a unit cell of the TMP, which consists of the front and rear sheets shown in (a). (c) System consisting of TMP-based metamaterials and rigid separators stacked vertically. Each layer consists of nine interlinked TMP unit cells (see Ref. [11] for details of such horizontal clustering). Conceptual illustrations of incident compressive waves and transmitted rarefaction waves are also shown.

Figure 2

(a) TMP unit cell. (b) Two-bar linkage model representing the folding motion of the two facets as marked in red lines in (a). (c) Force-displacement relationship of the TMP unit cell with $L=5\text{mm}$, $k_{\theta }=1.0\mathrm{Nm}/\mathrm{rad}$, and different initial folding angles: ${\theta }_{1,0}={30}^{\circ }$, ${45}^{\circ }$, and ${60}^{\circ }$. Dotted line indicates a power-law approximation of ${\theta }_{1,0}={45}^{\circ }$ case.

Figure 3

Schematic illustrations of (a) multibar linkage model and (b) lumped-mass model.

Figure 4

Space-time contour plots of strain-wave propagation based on (a) the multibar linkage model and (b) the lumped-mass model. Temporal plots of strain waves using (c) the multibar linkage model and (d) the lumped-mass model. Strain curves at $t=3$ ms and 40 ms are offset by 1.0 and 0.5, respectively, to ease visualization. The inset in (c) shows the magnified view of the leading edge. The arrows (1) and (2) point to the rarefaction wave present in the dynamics.

Figure 5

(a) Surface map of strain field to calculate wave speed. (b) Wave speed of strain waves as a function of external force ranging from $-150\text{N}$ to $+150\text{N}$. Numerical simulations are based on $L=25\text{mm}$, $m_j=19.7\text{g}$, $N=20$, and $k_{\theta }=1.0\mathrm{Nm}/\mathrm{rad}$.

Figure 6

Summary of numerical results on continuations of rarefaction waves over wave speed $c$ with $l=4001$ points and $\mathrm{\Delta }\xi =1/13$: (a) Relative strain profiles for various values of the wave speed $c$. (b) Relative momenta corresponding to (a). (c) Maximum of the absolute value of the relative strain variable as a function of the wave speed. Note that the horizontal dashed black line corresponds to the value of pre-compression in normalized units (or, equivalently, $d_0$ in physical units), while the vertical dashed-dot gray line corresponds to the value of the speed of sound $c_s$ of the medium.

Figure 7

Eigenvalue spectrum for a traveling wave with $c=180$ and $d_0=12\text{mm}$ on a lattice with $l=1501$ and $\mathrm{\Delta }\xi =0.2$.

Figure 8

Evolution of unperturbed (a)–(c) and perturbed (d)–(f) rarefaction waves with wave speed (a) and (d) $c=174$, (b) and (e) $c=180$, and (c) and (f) $c=190$, respectively, according to the discrete equation [Eq. (13)] with periodic boundary conditions. The parameter values used are the same as those described in the caption of Fig. 7.

Figure 9

Space-time contour plots of strain-wave propagation under tensile impact based on (a) the multibar linkage model and (b) the lumped-mass model. Temporal plots of strain waves using (c) the multibar linkage model and (d) the lumped-mass model. Strain curves at $t=3$ ms and 20 ms are offset by 1.0 and 0.5, respectively, to ease visualization. The inset in (c) shows the magnified view of the leading edge. The arrows (1) and (2) point to the solitary wave packets present in the dynamics.