Origami-based impact mitigation via rarefaction solitary wave creation

Abstract

The principles underlying the art of origami paper folding can be applied to design sophisticated metamaterials with unique mechanical properties. By exploiting the flat crease patterns that determine the dynamic folding and unfolding motion of origami, we are able to design an origami-based metamaterial that can form rarefaction solitary waves. Our analytical, numerical, and experimental results demonstrate that this rarefaction solitary wave overtakes initial compressive strain waves, thereby causing the latter part of the origami structure to feel tension first instead of compression under impact. This counterintuitive dynamic mechanism can be used to create a highly efficient-yet reusable-impact mitigating system without relying on material damping, plasticity, or fracture.

Fig. 1. Geometry of the TCO prototypes.

(A) Folding motion of the TCO is shown in sequence. (B) The flat sheet with crease patterns (upper left) is composed of mountain crease lines (red), valley crease lines (blue), and the adhesive area (shaded area). The photograph shows corresponding laser-cut paper sheets (lower right). (C) Actual prototype of the origami-based metamaterial and its unit cell (lower right inset). (D) The origami-based metamaterial generates the rarefaction solitary wave despite the application of compressive impact. The system is composed of the TCO unit cells (lower right). To connect the neighboring unit cells, we use the interfacial polygonal cross-section with markers at vertices (lower left). Photo credit: H.Y. and Y.M., University of Washington.

Fig. 2. Folding motions of the TCO with strain-softening behavior.

(A) The axial displacement (u) is defined with respect to the initial height (h₀) of the TCO. (B) Top-down view shows the rotational angle (ϕ) defined with respect to the initial angle (θ₀). (C) Axial force (F normalized by the spring constant K_a and h₀) versus displacement. The dashed red curve with the colored area represents the experimental value with the SD. The solid blue curve denotes the 2DOF linear spring model. The inset shows the variations of the stiffness as a function of the TCO displacement.

Fig. 3. Experimental setup and DIC analysis results.

(A) The shaker is attached to the leftmost unit cell through the sleeve bearing (upper left inset). The folding motion of each unit cell is captured by six action cameras (lower inset). For DIC analysis, the fluorescent green markers are used. (B) Snapshots of the experiment at t = 0, 0.06, 0.11, and 0.14 s. Images from the camera are shown in the left column, where the red (blue) arrows represent the compressive (tensile) velocity vector of the polygon in the axial direction. 3D reconstruction of the TCO chain (right column). The deformation is scaled 2.5 times larger than the original deformation for visual clarity. The gray arrows indicate the propagation of the rarefaction solitary wave. Photo credit: H.Y. and Y.M., University of Washington.

Fig. 4. Wave form analysis.

(A) Space-time evolution of the experimentally measured strain wave propagation in the origami-based system. The black arrow indicates the rarefaction solitary wave, and the green one shows the direction of the propagation. (B) Numerical simulation results show a qualitative agreement with the experimental data. The black arrow indicates the leading compressive wave in front of the rarefaction wave. (C) Amplitude change of the rarefaction solitary wave. The experimental data are fitted by the KdV solution (black curve) to obtain the damping coefficient for numerical and analytical analysis. The error bar represents the SD calculated from five measurements. Simulation results are shown in blue dots, which are fitted to the blue dashed curve. (D) The amplitude of the leading compression is analyzed. The dashed curves are obtained from the exponential fit to the experimental and numerical data. The inset shows the exponential decay of the compressive strain. The shapes of the rarefaction solitary wave (E) at t = 0.10 s and (F) t = 0.15 s are shown.

Fig. 5. Wave speed analysis.

(A) Space-time contour plot of the strain wave for the numerical simulation conducted on the longer chain composed of 50 TCO unit cells. Magnified view of the overtaking moment is shown in the right inset. (B) Trajectory of the rarefaction solitary wave (denoted by the blue markers) and the maximum compressive strain wave (red markers) shows the overtaking behavior of the rarefaction solitary wave. The green line indicates the analytical prediction from the KdV equation. (C) Wave speed of the rarefaction solitary wave is higher than the speed of sound of the medium, which means supersonic behavior.