SCIENCE ADVANCES | RESEARCH ARTICLE
ENGINEERING

Origami-based impact mitigation via rarefaction
solitary wave creation
Hiromi Yasuda1, Yasuhiro Miyazawa1,2, Efstathios G. Charalampidis3, Christopher Chong4,
Panayotis G. Kevrekidis3, Jinkyu Yang1*
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.

the TCO is characterized by coupling between axial and rotationalmotions, as shown in Fig. 1A. Recent studies on the TCO structure haveshown their versatile mechanical properties, such as its tailorable stability (20), zero-stiffness mode (31), and strain-softening/hardening behavior (21).
Our origami structure consists of 20 TCO unit cells, which are fabricated by using paper sheets cut by a laser cutting machine (Fig. 1, B
and C; see also Materials and Methods for details). The crease patterns
are carefully designed to make the TCO cells exhibit effective strainsoftening behavior, as in (21). This softening behavior in origami isin sharp contrast to conventional nonlinear metamaterials that mainly
exploit strain-hardening behavior, such as granular crystals (8, 11, 32),that have been serving as a popular test bed for demonstrating classicaldynamics of nonlinear mass-spring systems [e.g., the celebrated FermiPasta-Ulam-Tsingou chain (33)].
Recently, it has been theoretically and numerically predicted thata 1D discrete system with strain-softening interactions can support
the propagation of a solitary wave in the form of a rarefaction pulse(8, 34, 35). In this 1D system, the application of a compressive impact
generates a tensile solitary wave, propagating ahead of the initial compressive strain wave and thereby making the latter part of the medium
feel tension first instead of compression (see Fig. 1D for the conceptual
illustration). While the feasibility of verifying this counterintuitive
dynamics has been discussed in different settings (35, 36), the experimental demonstration of these rarefaction solitary waves remains
elusive. This is mainly due to the challenges in fabricating an effective
strain-softening medium that exhibits minimal dissipation for waveguiding purposes.
Here, we report the first experimental observation of a mechanicalrarefaction solitary wave in the strain-softening setting of TCO-based
metamaterials. This rarefaction wave overtakes the leading compressive
wave generated by a compressive impact. The experimentally observed
rarefaction pulse agrees quantitatively with numerical simulations of thederived model and agrees qualitatively with an asymptotic analysis
based on the celebrated Korteweg–de Vries (KdV) equation (37, 38).

1

Department of Aeronautics & Astronautics, University of Washington, Seattle, WA
98195-2400, USA. 2Department of Mechanical and Aerospace Engineering, Tohoku
University, Sendai, Miyagi 980-8577, Japan. 3Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA. 4Department ofMathematics, Bowdoin College, Brunswick, ME 04011, USA.
*Corresponding author. Email: jkyang@aa.washington.edu
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STATIC ANALYSIS

We start by examining the geometry of the TCO unit cell. The TCO
structure has axial and rotational motions that are coupled to eachother. We characterize this folding motion by using the axial displacement
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INTRODUCTION

Mechanical metamaterials offer a new dimension in achieving nonconventional and tailored mechanical properties through architecture(1–3). In particular, the manipulation of wave propagation in mechanical metamaterials is a topic of intense research for various engineering applications, e.g., waveguiding, vibration filtering, subwavelength
imaging, and impact mitigation (4–7). The ability to achieve this desirable mechanical performance often relies on the platform in which weconstruct mechanical metamaterials. The choice of platform can varyfrom periodically arranged microlattice/macrolattice structures toself-assembling particles to three-dimensional (3D) printed soft/hardarchitected materials (8–11).
Recent studies have shown that origami can serve as an ideal playground to realize highly versatile and tunable mechanical metamaterials(12–16). For example, by introducing crease lines into flat surface
materials, one can construct origami-based structures, which can offer
enhanced stiffness (17), negative Poisson’s ratio (18), and multistability(19–21). Given the scale-free nature of origami, this kind of design
framework can be used in a wide range of scales. For example, the concept of origami has been adapted to a diverse set of design principles,
including robotics (22), reconfigurable structures (23), and self-folding
actuated by living cells (24). While these studies focused mainly on origami’s static or quasi-static behavior, the analysis of the dynamics oforigami-based structures is a natural next step to investigate. However,the connection between the origami crease pattern and the dynamic
folding/unfolding behavior of origami itself has been relatively unexplored (25, 26). In particular, very few experimental studies have beenreported (27, 28).
In the present study, we explore unique wave dynamics in a mechanical metamaterial that is composed of volumetric origami structures.
In particular, each volumetric origami structure is a triangulated cylindrical origami (TCO) (Fig. 1A for the folding motion and Fig. 1B for theflat crease pattern before it is assembled into a unit cell) (21). This TCO
unit cell is analogous to a post-buckled shape of a cylinder under simultaneous axial and twisting loading (29, 30). Thus, the folding motion of

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SCIENCE ADVANCES | RESEARCH ARTICLE
A

B

C

D

(u) and rotational angle (φ), which are defined with respect to the initialheight (h0) and angle (q0) (Fig. 2 , A and B). To analyze the kinematics ofthe origami, we construct a two–degree of freedom (2DOF) mathematical model, which approximates the folding behavior of the TCO
into linear spring motions along the crease lines (21). In this model,
we use two different spring constants (Ka and Kb) for the shorter(Aa B in Fig. 2A) and longer crease lines (Ab B), respectively. These twolinear springs are repeated along the Np-sided polygon (denoted by theshaded area with the radius R of the circle circumscribing the polygon). In addition, the side crease (Aa Ab) is modeled as a torsional spring
to capture the folding of the side facets with the angle of y (Fig. 2A). The
torsional spring constant for this crease line is denoted by Ky. These
three spring constants are determined empirically by conducting compression tests on the fabricated TCO cells (see Materials and Methods
for details and movie S1 for the fabrication and folding motion of
the TCO).
We analyze the elastic potential energy change of the TCO unit
cell as a function of u and φ as follows

A

Top
B

y

B

Bottom
Top

O
R

Ab

C

Bottom
Aa

Ab

x
B

Aa

2DOF model11Uðu; φÞ ¼ Np Ka ða $ að0Þ Þ2 þ Np Kb ðb $ bð0Þ Þ2 þ
2
2
1
ð2Np ÞKy ðy $ yð0Þ Þ22

ð1Þ

where a = a(u, φ) and b = b(u, φ) are the length of the crease lines (Aa B
and Ab B), respectively, and the superscript (0) denotes the initial states.
By using this energy expression and applying the minimum potential
energy principle (21), we obtain the 2DOF model that indicates how the
deformation of the TCO cell takes place by coupling axial and rotational
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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) Topdown view shows the rotational angle (ϕ) defined with respect to the initial angle (q0).
(C) Axial force (F normalized by the spring constant Ka 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 thevariations of the stiffness as a function of the TCO displacement.
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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 mountaincrease 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 ofthe 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 atvertices (lower left). Photo credit: H.Y. and Y.M., University of Washington.

SCIENCE ADVANCES | RESEARCH ARTICLE
motions (see the “Equations of motion” section in the Supplementary
Materials for details). Figure 2C shows the analytical prediction (solid
curve) obtained from this 2DOF model, along with the experimental
force and displacement relationship (dashed curve). See Materials
and Methods for the detailed testing and table S1 for the parameters
used in experiments and analytics. We observe the stiffness of thestructure decreases as it is compressed, while the trend is opposite
under tension (inset of Fig. 2C). We will later discuss how thisstrain-softening behavior contributes to the unique wave dynamics
in this TCO-based system.
DYNAMIC ANALYSIS

A

B

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 propagationof the rarefaction solitary wave. Photo credit: H.Y. and Y.M., University of Washington.
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With a firm understanding of the unit cell at hand, we are ready toinvestigate the wave dynamics in a chain of TCO cells (Fig. 3A).
For experiments, we prototype 20 identical TCO unit cells by choosing
h0 = 35 mm, q0 = 70∘, R = 36 mm, and Np = 6. To ensure uniform andrepeatable folding behavior in these units, we apply the preconditioning
process to each cell and verify the uniformity by measuring its compressive motions (see Materials and Methods for details). The left end of thechain (unit number n = 1) is connected to a shaker through the customized attachment with a sleeve bearing, which transfers the shaker
impact to the cell while allowing its free rotational motion (see the upper
inset of Fig. 3A). The TCO cell positioned in the right end of the chain(n = 20) is fixed to the rigid wall. To measure the dynamic folding/unfolding motion of each unit cell, we use the digital image correlation(DIC) technique by using three pairs of action cameras (GoPro HERO4

Black) whose maximum frame rate is 240 frames per second (see thelower inset of Fig. 3A).
The digital images in Fig. 3B show the snapshots of the first eight
TCO unit cells at four different time frames captured by the first pairof the action cameras (see movie S2). In this figure, the length of thecolored arrows represents the speed of the polygon in the axial direction,and red (blue) color denotes rightward (leftward) velocity. For the sake ofvisualization, the 3D images are reconstructed based on the experimental
data as shown in the right column of Fig. 3B, where red (blue) color
indicates compressive (tensile) strain (see movie S3). Here, the strain
value of the nth cell is defined by (un − un + 1)/h0, where un (un + 1) isthe axial displacement of its left (right) polygon such that compressive
strains take positive signs for convenience. From the experimental
results, we observe that, initially, the first unit shows the large-amplitude
compression due to the excitation by the shaker. However, this compressive motion decays quickly without being robustly transmitted
along the chain, whereas the noticeable tensile motion is evolved instead (see the gray arrows in Fig. 3B and also movie S2). Thus, it appears that a tensile wave has propagated despite the application of acompressive force.
To conduct a more thorough analysis of this counterintuitive wave
dynamics, we plot the measured strains in time and space domains (Fig.
4A). We observe evidently that the last TCO cell (n = 20) experiences atensile strain (see the black arrow) despite the application of compressive impact to the system. This tensile strain is due to the formation andpropagation of a rarefaction solitary wave (blue valley as indicated by
the green arrow). Rarefaction (i.e., tensile) solitary waves are the conical

SCIENCE ADVANCES | RESEARCH ARTICLE
B

C

D

E

F

Fig. 4. Wave form analysis. (A) Space-time evolution of the experimentally measured strain wave propagation in the origami-based system. The black arrow indicatesthe rarefaction solitary wave, and the green one shows the direction of the propagation. (B) Numerical simulation results show a qualitative agreement with theexperimental 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 theSD 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 thecompressive strain. The shapes of the rarefaction solitary wave (E) at t = 0.10 s and (F) t = 0.15 s are shown.

type of solitary wave in strain-softening systems (34), such as the TCO
chain considered herein. One would not expect to find compressive solitary waves in our system because they are observed typically in strainhardening systems, such as granular crystals (8).
To better understand the above experimental results, we formulate
the equations of motion for the entire chain based on the 2DOF modelof the TCO cells and solve the resulting equations numerically by using
the fourth-order Runge-Kutta method (see the “Equations of motion”
section in the Supplementary Materials for details). The simulation
results are shown in Fig. 4B, which demonstrates excellent agreementwith the experimental results. The dissimilar trend of wave propagation
between the tensile and compressive waves in our system is worth
noting (compare the blue and red peaks indicated by black arrows in
Fig. 4, A and B). To examine the difference between these two types
Yasuda et al., Sci. Adv. 2019; 5 : eaau2835

24 May 2019of waves, we extract and compare the strains of the leading tensileand compressive waves in Fig. 4, C and D, respectively. We find thatthe compressive component decays more drastically than the tensile
counterpart. The compressive waves show an order-of-magnitude reduction in amplitude, as verified by the exponential decay of the compressive strain in the inset of Fig. 4D.
The strong contrast of wave attenuation trends stems from the intrinsic nature of the two different waves formed in the tensile and compressive realms. In the tensile domain, our TCO system generates
rarefaction solitary waves in a robust form, while in the compressive
region, it forms oscillatory waves that disperse energy into space andtime (see oscillations in Fig. 4, A and B). As will be verified later, therarefaction solitary waves are supersonic and thus nonlinear, while
the oscillatory ones are linear and limited by the sound speed. These
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SCIENCE ADVANCES | RESEARCH ARTICLE
plementary Materials). This equation has a closed-form rarefaction solitary wave solution, thereby yielding an analytical approximation of the
wave in the TCO lattice. Note that this analytical approach captures
the effect of damping (see the gradual decrease of the KdV curve in
Fig. 4C) based on the empirical dashpot model (see the “Equation of motion with damping effect” and “Continuum limit” sections in the Supplementary Materials).
By using the experimental, numerical, and analytical data, we now
investigate the evolution of the strain waves at t = 0.10 and 0.15 s, as
shown in Fig. 4 (E and F). In these plots, we already witness a substantially attenuated form of the leading compressive wave. On the contrary,
the tensile one is propagating more dominantly and robustly. Although
the analytical waveform focuses only on the tensile components, we observe a qualitative agreement among analytical, numerical, and experimental results. One interesting finding here is that the maximum
compressive strain is ahead of the rarefaction solitary wave at t = 0.10 s,
but the peak is shifted to the rear part of the solitary wave at t = 0.15 s.
This indicates that the rarefaction solitary wave propagates faster thanthe other compressive wave packets and eventually overtakes the initial
compressive strain wave. While this overtaking behavior has been previously studied numerically (34, 35), this is the first experimental observation in mechanical platforms to the best of our knowledge.
We now further investigate the overtaking behavior by conducting
numerical simulations for a longer chain composed of 50 unit cells (seeFig. 5A and its inset for the magnified view). Upon the compressive
impact, the TCO chain initially experiences a compression, followed

A

B

C

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.
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dissimilar mechanisms are attributed to the strain-softening nature of
the TCO system in compression, where the cell stiffness decreases (increases) as a further compression (tension) is imposed (see the inset of
Fig. 2C). In nonlinear mass-spring systems, this asymmetric behavior
implies that the system will favor wave localization in tension and dispersion in compression (34).
Although general strain-softening systems can also be used for
impact mitigation purposes, they typically require hundreds of cells
to generate rarefaction solitary waves based on previous numericalstudies (34–36). In the present experimental work, however, we findthat our TCO architecture can form rarefaction waves even within
10 TCO cells by leveraging the tailorable strain-softening nature of the
TCO and its unique wave-coupling motions. This manifests the efficacyof the TCO-based metamaterial in mitigating the compressive impact.
The robust propagation of the leading tensile waves can also bemodeled by theoretical analysis. To this end, we first postulate un~φnbased on the eigenmode of the single TCO unit cell. Note that this linear
relationship holds approximately in the full range of dynamic strainsconsidered here (see the “Derivation of single-component model” section in the Supplementary Materials for an a posteriori validation of thisapproximation). On the basis of this, the equations of motion can be
reduced to a single-component model. Note that the single-component
model is still nonlinear by virtue of the force-displacement andtorque-angle relationships. Then, we take the continuum limit of thissingle-component equation in the infinite TCO chain to derive thewell-known KdV equation (see the “Continuum limit” section in the Sup-SCIENCE ADVANCES | RESEARCH ARTICLE

CONCLUSION

We have studied experimentally, numerically, and analytically a remarkable example of nonlinear wave propagation in mechanicalmetamaterials made of volumetric origami cells. We found that the
TCO-based metamaterials exhibit the rarefaction solitary wave, whichfeatures tensile strains and propagates ahead of the initial compressive
strain despite the application of external compressive impact. The KdVbased theoretical analysis provided us with an excellent qualitative
handle for understanding the relevant dynamics, and the numericalcomputations corroborated the experimental observations. Also, theinitial compressive strain is attenuated significantly, which can behighly beneficial for impact mitigation applications. While we focusedon the monoatomic TCO unit cells with strain-softening behavior inthis study, the origami-based system has great potential for supportingrich wave dynamics by introducing heterogeneous elements (e.g.,hardening, multistable, and impurity components) in the chain. Also,
the findings in this 1D setting can be further extended to multidimensions in a modular way. We believe that this architecture of volumetricorigami cells can be used as a versatile building block for a wide range
of applications such as impact/shock mitigation, vibration filtering,and energy harvesting.
MATERIALS AND METHODS

Prototype fabrication
We used construction paper sheets (Strathmore 500 Series 3-PLY
BRISTOL; thickness of the paper is 0.5 mm) for the origami part(see Fig. 1B) and extruded acrylic sheets (thickness of 1.6 mm) for theinterfacial polygon (insets of Fig. 1, C and D). These two materials were
tailored by a laser cutting machine (VLS 4.6, Universal Laser Systems),
Yasuda et al., Sci. Adv. 2019; 5 : eaau2835

24 May 2019as shown in fig. S1A. For the crease of the TCO unit cell, we designed
customized crease lines based on the compliant mechanisms (fig.
S1B). By using this crease pattern, each crease line shows enhanced
fatigue-resistant property, providing repeatable and consistent folding
behavior. It should also be noted that this crease pattern (e.g., length,
width, and distance of cut slits) affects the compressive forcedisplacement behavior of the TCO cell significantly. The weight andmoment of inertia of the interfacial polygon including a sleeve bearing
and markers are 49.4 g and 5.32 ×10−5 kg m2, respectively. The weight of
the triangulated facets made of paper is only 3.6 g. Therefore, the inertiaof the side facets is negligible compared to the polygon. The assemblingprocess of a single TCO cell is shown in movie S1.
Compression test on the TCO unit cell
The static compression was conducted in a customized testing
environment. The unit cell prototype was placed vertically on the load
frame (fig. S2), where the bottom part of the unit cell was fixed. The top
end of the unit cell was connected to the load cell (LUX-B-50 N-ID,
Kyowa) through a sleeve bearing made of polytetrafluoroethylene(PTFE) so that the top polygon was constrained in the axial direction
but could rotate freely around the longitudinal axis. The top part holding the load cell was actuated by the motor-driven linear stage (BiSlider,
Velmex). The displacement was measured by a noncontact laser sensor(CMOS Multi-Function Analog Laser Sensor IL-065, Keyence).
Preconditioning process on the TCO unit cell
Each TCO prototype was assembled by hand; therefore, the folding behavior, specifically force-displacement relationship, varies among thesamples. Similarly, the repeatability of folding/unfolding behavior ofeach unit cell is also a critical factor to ensure reliable performance of
the whole origami system. To address this issue, we conducted fatigue
tests on all unit cells by using the load frame explained above. We applied 200 cycles of controlled displacement from −3 mm (tension) to
15 mm (compression) at 6 mm/s, and we measured the corresponding
force-displacement relationship. Figure S3A shows the loading and unloading curves from the first cycle (i.e., newly fabricated state without
experiencing axial force) to the 200th cycle. One of the noticeablefeatures is that the first cycle shows the significant maximum force peakduring loading (denoted by the black curve in fig. S3A). Then, as thenumber of cycles increases, each curve traces a similar path.
To analyze this fatigue behavior, we considered the area bounded by
the loading and unloading curves for each hysteresis loop, and weplotted the area as a function of the number of cycles as shown in fig.
S3B. The area enclosed by the hysteresis loop is associated with energy
dissipation. For the initial 50 cycles, the energy dissipation decreases significantly, and then the slope of the energy dissipation curve acquires a
relatively small value. Hence, to ensure homogeneous and repeatablecharacteristics of the TCO chain for the dynamic test, we applied thiscyclic preconditioning process (200 cycles, displacement of −3 to 15 mm)
to all unit cells and selected those with highly consistent and uniform
folding behavior.
Dynamic test on the chain of the TCO unit cells
We conducted the dynamic test on the chain of the TCO unit cells byapplying compressive impact to the system. For the chain of the TCO
unit cells (Fig. 1C), neighboring unit cells were connected mechanicallyby using M3 stainless steel screws and hex nuts. Also, a flanged sleevebearing made of PTFE was embedded in the center of the interfacial
polygon, through which the stainless steel shaft (diameter of 4.76 mm)
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by the tension resulting from the reaction of the TCO cells. In this case,the speed of the initial (i.e., leading) compressive wave is bounded by thesystem’s sound speed due to its effectively linear nature. However, thefollowing rarefaction pulse is predicted to be supersonic by the KdV
analysis. Thus, we would expect the rarefaction pulse to overtake thecompressive linear waves (see the “Overtaking behavior” section in
the Supplementary Materials for details). We observe this overtaking
phenomenon around the fifth cell location (see inset of Fig. 5A), thereby
verifying the higher speed of the rarefaction solitary wave than that of
the compressive wave pattern.
To quantify the speeds, we extract the time and location of the maximum peak of the rarefaction solitary and compressive strain waves, as
shown in Fig. 5B. As marked by the circle in the figure, we detect theovertaking moment around t = 0.11 s when the trajectories of solitary
wave and maximum compression peak are crossing. The slope of eachcurve represents the wave speed; therefore, the wave speed of the rarefaction solitary waves can be calculated from the experimental and numerical results. Figure 5C shows the wave speed calculated from theexperiment (20 units) and simulation (50 units), and the result for
the rarefaction solitary wave is compared with the prediction fromthe KdV solution (see the “Continuum limit” section in the Supplementary Materials for the details about the KdV and sound speed calculations). We observe that the experimental and numerical results
corroborate the analytical prediction, confirming the supersonic natureof the rarefaction pulse (see the “Overtaking behavior” section in the Supplementary Materials for more details about the overtaking mechanisms
in various scenarios).

SCIENCE ADVANCES | RESEARCH ARTICLE

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/
content/full/5/5/eaau2835/DC1
Supplementary Text
Fig. S1. Fabrication of the TCO unit cell.
Fig. S2. Compression test on the TCO unit cell.
Fig. S3. Fatigue property of the TCO single unit cell.
Fig. S4. DIC technique to measure the axial displacement and the rotational angle of eachpolygon.
Fig. S5. Surface plot of the elastic potential energy.
Fig. S6. Effect of damping factor n.
Fig. S7. Comparison between the numerical simulations on the TCO chain with/without
damping effect.
Fig. S8. Comparison between the numerical simulations on the nonlinear/linearized TCO chain.
Fig. S9. Numerical simulation with the application of the initial impact velocity to the first unit.
Table S1. Numerical constants used in the numerical simulation and analytics.
Movie S1. Fabrication of the TCO unit cell.
Movie S2. Experimental demonstration of the rarefaction solitary wave.
Movie S3. 3D reconstruction of the TCO chain from the experimental result.

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was inserted to align the unit cell. In this way, a chain of TCO cells
will have free axial and rotational motions while restricting thebending of the chain.
The left end of the chain (first unit cell) was connected to the shaker(LDS V406 M4-CE, Brüel & Kjær) through the customized attachment(see the inset of Fig. 3A) with a sleeve bearing (PTFE), which attaches
the leftmost polygon of the cell to the shaker but allows free rotationalmotions. The right end of the chain (20th unit cell) was fixed to the rigid
wall. The shaker was excited by a single-step voltage to apply compressive impact to the system.
To capture the dynamic folding/unfolding motion of the chain, wedeveloped a customized noncontact DIC technique by using Python. To
track the motion of each interfacial polygon, we used the color andshape as the features of a target marker. We attached a spherical marker
to each corner of the polygons. For the color of the marker, fluorescentgreen was used to distinguish the marker of the polygons from the otherobjects. Figure S4A is the original image from the GoPro camera. With
the mask based on the fluorescent green, we extracted the marker color
area from the raw digital image (see the lower inset of fig. S4A). Last, we
identified the marker from this filtered image by examining the shape ofthe extracted area and determined the 3D coordinate of each marker
based on the triangulation method. By using three pairs of the action
cameras, we split the field of view horizontally by three and captured
the axial and rotational motions of the polygons along the longitudinal
axis based on the stereo vision. The six cameras were calibrated before
the measurements, and our in-house code processed the rectified
images to obtain the 3D coordinate information. Figure S4B shows
the axial displacement change of the first interfacial polygon mounted
on the shaker attachment. For numerical simulations, we fed this experimentally obtained displacement data into the equation of the first TCO
element’s motion.

SCIENCE ADVANCES | RESEARCH ARTICLE
37. M. J. Ablowitz, Nonlinear Dispersive Waves (Cambridge Univ. Press, 2011).
38. R. Khomeriki, Interaction of a kink soliton with a breather in a Fermi-Pasta-Ulam chain.
Phys. Rev. E 65, 026605 (2002).
Acknowledgments: We thank S.-g. Ahn and F. O’Brien of the School of Art + Art History +Design at the University of Washington for technical support. J.Y. and P.G.K. also acknowledge
the support from the AFOSR (FA9550-17-1-0114). Funding: This material is based on work
supported by the NSF under grant no. CAREER-1553202 (J.Y.), DMS-1615037 (C.C.), andDMS-1809074 (P.G.K.). J.Y. and H.Y. are grateful for the support from the ONR (N000141410388)
and the Washington Research Foundation. Author contributions: H.Y. and J.Y. conceived
the research topic. H.Y. and Y.M. conducted the design, fabrication, and testing of the prototypes.
H.Y. conducted the numerical simulations. H.Y., E.G.C., C.C., and P.G.K. created the analyticalmodel and interpreted the computational results. J.Y., C.C., and P.G.K. provided guidance throughout
the research. All authors contributed to writing and editing the manuscript. Competing interests:

H.Y. and J.Y. are inventors on a patent application related to this work filed by the University ofWashington (patent application no. 62/562,596, filed on 25 September 2017). The other authors
declare that they have no competing interests. Data and materials availability: All data needed
to evaluate the conclusions in the paper are present in the paper and/or the Supplementary
Materials. Additional data related to this paper may be requested from the authors.
Submitted 24 May 2018
Accepted 12 April 2019
Published 24 May 201910.1126/sciadv.aau2835
Citation: H. Yasuda, Y. Miyazawa, E. G. Charalampidis, C. Chong, P. G. Kevrekidis, J. Yang,
Origami-based impact mitigation via rarefaction solitary wave creation. Sci. Adv. 5, eaau2835(2019).

Downloaded from http://advances.sciencemag.org/ on November 8, 2020

Yasuda et al., Sci. Adv. 2019; 5 : eaau2835

24 May 20198 of 8

Origami-based impact mitigation via rarefaction solitary wave creation
Hiromi Yasuda, Yasuhiro Miyazawa, Efstathios G. Charalampidis, Christopher Chong, Panayotis G. Kevrekidis and Jinkyu
Yang

Sci Adv 5 (5), eaau2835.
DOI: 10.1126/sciadv.aau2835

http://advances.sciencemag.org/content/5/5/eaau2835

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