Metamaterials with amplitude gaps for elastic solitons

Abstract

We combine experimental, numerical, and analytical tools to design highly nonlinear mechanical metamaterials that exhibit a new phenomenon: gaps in amplitude for elastic vector solitons (i.e., ranges in amplitude where elastic soliton propagation is forbidden). Such gaps are fundamentally different from the spectral gaps in frequency typically observed in linear phononic crystals and acoustic metamaterials and are induced by the lack of strong coupling between the two polarizations of the vector soliton. We show that the amplitude gaps are a robust feature of our system and that their width can be controlled both by varying the structural properties of the units and by breaking the symmetry in the underlying geometry. Moreover, we demonstrate that amplitude gaps provide new opportunities to manipulate highly nonlinear elastic pulses, as demonstrated by the designed soliton splitters and diodes.

Full text

ARTICLE
Metamaterials with amplitude gaps for elastic
solitons
Bolei Deng1, Pai Wang1,QiHe
2, Vincent Tournat 3& Katia Bertoldi1,4
We combine experimental, numerical, and analytical tools to design highly nonlinear
mechanical metamaterials that exhibit a new phenomenon: gaps in amplitude for elastic
vector solitons (i.e., ranges in amplitude where elastic soliton propagation is forbidden). Such
gaps are fundamentally different from the spectral gaps in frequency typically observed in
linear phononic crystals and acoustic metamaterials and are induced by the lack of strong
coupling between the two polarizations of the vector soliton. We show that the amplitude
gaps are a robust feature of our system and that their width can be controlled both by varying
the structural properties of the units and by breaking the symmetry in the underlying geo-
metry. Moreover, we demonstrate that amplitude gaps provide new opportunities to
manipulate highly nonlinear elastic pulses, as demonstrated by the designed soliton splitters
and diodes.
DOI: 10.1038/s41467-018-05908-9 OPEN
1Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. 2School of Aerospace Engineering,
Tsinghua University, 100084 Beijing, China. 3LAUM, CNRS, Le Mans Université, Av. O. Messiaen, 72085 Le Mans, France. 4Kavli Institute, Harvard
University, Cambridge, MA 02138, USA. Correspondence and requests for materials should be addressed to K.B. (email: bertoldi@seas.harvard.edu)
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Following John Scott Russell’s observation of nonlinear water
wave packets propagating with stable shape and constant
velocity in the Union Canal in Scotland1, the unique
properties of solitons have been studied and exploited in many
areas of science and engineering2–4. Focusing on mechanical
systems, granular crystals have been found to provide an effective
platform for the propagation of highly nonlinear solitary waves4–7
and have enabled the design of impact mitigation layers8, lenses9,
switches10, and non-destructive detection techniques11. However,
the solitons observed in granular media are of scalar nature and
lack the multiple polarizations typical of elastic waves propagat-
ing in solid materials.
Polarization is an important property of vector waves like
electromagnetic and elastic waves. The ability to control the
polarization of light has enabled a broad range of applications,
including optical communications, spectroscopy, and micro-
scopy3,12,13. Moreover, a broad range of new functionality has
been observed in elastic systems with architecture designed to
manipulate both the longitudinal and shear polarizations of linear
elastic waves14,15. While the field initially focused on linear elastic
vibrations, it has been recently shown that highly deformable
mechanical metamaterials can support elastic vector solitons with
two polarizations—one translational and one rotational16, but the
potential of such solitary waves in applications is unknown and
remains to be explored.
Here, we combine experimental, numerical, and analytical
tools to demonstrate that elastic vector solitons provide unique
opportunities to manipulate the propagation of large amplitude
vibrations. Specifically, we show that in mechanical metamaterials
based on rotating rigid units, both the amplitude of the propa-
gating waves and the symmetry of the underlying building blocks
can be used to significantly alter the coupling between the two
polarizational components of the vector solitons. We find that
such control of the coupling strength results in the emergence of a
new phenomenon: the formation of amplitude gaps for solitons.
Notably, this new effect can be exploited to realize devices capable
of controlling and manipulating the propagation of large ampli-
tude vibrations in unprecedented ways, as demonstrated by the
design of soliton splitters and diodes.
Results
Metamaterial design and characterization. Our system consists
of a long chain of 2 × 50 rigid crosses made of LEGO bricks17
with arm length l
a
=19 mm connected by thin and flexible hinges
made of polyester plastic sheets (Artus Corporation, NJ) with
length l
h
=4 mm and thickness t
h
=0.127 mm, resulting in a
spatial periodicity a=2l
a
+l
h
=42 mm (Fig. 1a—see Supple-
mentary Note 1 and Supplementary Movie 1 for details on fab-
rication). To investigate the propagation of elastic pulses in the
system, we place the chain (supported by pins to minimize fric-
tions) on a smooth horizontal surface and use an impactor
excited by a pendulum to hit the mid-point at its left end (see
Fig. 1b-top and Supplementary Movie 2). We apply different
input signals to the chain by varying both the initial height of the
striking pendulum and the distance traveled by the impactor and
find that all of them initiate simultaneous rotation and dis-
placement of the rigid units, with each pair of crosses in a column
sharing the same displacement and rotating by the same amount,
but in opposite directions (i.e., if the top unit rotates by a certain
amount in clockwise direction, then the bottom one rotates by the
same amount in counter-clockwise direction, and vice versa). To
monitor the displacement, u
i
, and rotation, θ
i
, of the i-th pair of
crosses along the chain as the pulse propagates, we use a high
speed camera (SONY RX100V) and track markers via digital
image processing (see Supplementary Note 2).
In Fig. 1c, we report the evolution of the rotation and
longitudinal displacement of the second and fortieth pairs of
crosses as a function of time during two different experiments.
We find that when the amplitude of the input signal is large (A
2
=max(θ
2
(t)) =13° in experiment #1) the pulse that propagates
through the system conserves its amplitude and shape in both
degrees of freedom. Differently, for inputs with small amplitude
(A
2
=5° in experiment #2), the output signal is severely distorted
compared to the input one (see Supplementary Movie 3). While
in Fig. 1c we focus on two representative experiments, all our
experimental results are summarized as triangular markers in
Fig. 1e, f, where we present the measured transmission, A
40
/A
2
(with A
40
=max(θ
40
(t))), and cross-correlation of θ
2
(t) and θ
40
(t)
as a function of the amplitude of the input signal, A
2
.Wefind
that if A
2
≳7° both the transmission and the cross-correlation
approach unity, suggesting that for large enough input signals, the
system supports the propagation of elastic vector solitons.
However, for input amplitudes below ~7° a transition occurs
and both the transmission and the cross-correlation significantly
and systematically decrease. This indicates that our system might
only support the propagation of solitary waves with amplitude
above a certain threshold, manifesting a gap in amplitude for
0° ≲A
2
≲7°.
Discrete and continuum models. To better understand these
experimental results, we establish a discrete model in which the
crosses are represented as rigid bodies of mass mand rotational
inertia J. Guided by our experiments, we assume that the system
has an horizontal line of symmetry and assign two degrees of
freedom (u
i
and θ
i
) to the top unit of the i-th pair of crosses
(Fig. 1d). As for the flexible hinges, they are modeled using a
combination of three linear springs: their stretching is captured
by a spring with stiffness k
l
; their shearing is described by a spring
with stiffness k
s
; their bending is modeled by a torsional spring
with stiffness k
θ
. Under these assumptions, the dimensionless
equations of motion for the i-th top unit are given by (see Sup-
plementary Note 3)
∂2Ui
∂T2¼Uiþ12UiþUi1cosθiþ1þcosθi1;
1
α2
∂2θi
∂T2¼Kθθiþ1þ4θiþθi1

þKscosθisinθiþ1þsin θi12 sin θi

sinθi2Uiþ1Ui1

þ4cosθiþ1

2cosθicosθi1;
ð1Þ
where U
i
=u
i
/a,T=tffiffiffiffiffiffiffiffiffiffi
kl=m
p,K
θ
=4kθ=kla2
ðÞ,K
s
=k
s
/k
l
and α
=affiffiffiffiffiffiffiffiffiffiffiffiffiffi
m=ð4JÞ
pare all non-dimensional parameters, which in our
system are measured as K
s
=0.02, K
θ
=1.5 × 10−4, and α=1.8.
Note that, as shown in Fig. 1d, to facilitate the analysis in our
model we define the positive direction of rotation alternatively for
neighboring crosses (i.e., if for the i-th top unit a clockwise
rotation is positive, then for the (i−1)-th and (i+1)-th ones
counterclockwise rotation is positive).
We start by numerically solving Eq. (1) using the Runge–Kutta
method (the code implemented in MATLAB is available online).
In our numerical analysis we consider a chain comprising 150
pairs of crosses, apply a longitudinal displacement with the form
U
input
=b+btanh[(T−T
0
)/w] to the mid-point at its left end
(see Supplementary Note 3), and implement free-boundary
conditions at its right end. In Fig. 1e, f we report as gray circular
markers the results of 480 analyses in which we systematically
change the applied displacement U
input
(with b∈[0, 0.75], w∈
[50, 100] and T
0
=400). In good agreement with our experi-
mental data, we find that for A
2
≲6° the signal does not preserve
its amplitude and shape as it propagates through the structure. As
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such, these numerical results also point to the existence of an
amplitude gap for solitons.
Amplitude gaps for solitons. To confirm the existence of such
amplitude gap, we further simplify Eq. (1) to obtain an analytical
solution. To this end, we assume that the wavelength of the
propagating waves is much wider than the cell size and that
θ1, take the continuum limit of Eq. (1) and retain nonlinear
terms up to the third order, obtaining (see Supplementary Note 4)
∂2U
∂T2¼∂2U
∂X2þθ∂θ
∂X;
1
α2
∂2θ
∂T2¼KsKθ
ðÞ
∂2θ
∂X243Kθ
2þ∂U
∂X
hi
θ2θ3;ð2Þ
where X=x/a(xdenoting the initial position along the chain)
and U(X,T) and θ(X,T) are two continuous functions of Xand T.
It is easy to show that Eq. (2) admits an analytical solution in the
form of an elastic vector soliton with two components18
θ¼Asech XcT
W

;
U¼A2W
21c2
ðÞ
1tanh XcT
W

;
(ð3Þ
where cis the pulse velocity, and Aand Ware the amplitude and
width of the solitary wave, which can be expressed in terms of c
and the structural parameters as (see Supplementary Note 4)
A¼±ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6Kθ1c2
ðÞ
c2
r;and W¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
α2KsKθ
ðÞc2
6α2Kθ
s:ð4Þ
At this point it is important to note that, since the width Wneeds
to be real-valued
c2<α2KsKθ
ðÞ;ð5Þ
yielding
A
jj
>Aupper ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6Kθ
α2KsKθ
ðÞ
6Kθ
s:ð6Þ
Condition in Eq. (6) clearly indicates that our system has an
amplitude gap for solitons, since only solitary waves with
amplitude greater than A
upper
=6.55° are physically admissible
0 0.1 0.2
Velocity c
0 0.5 1 –1 10
bInput Output
i = 2
i = 2
i = 2
i = 40
i = 40
i = 40
e
00.1
Time (s)
Angle i
Angle i
i
i
i + 1
i + 1
Disp. ui (mn)
Disp. ui (mn)
00.1
Time (s)
0
2
4
6
8
10
0 0.05 0.1
Time (s)
0 0.05 0.1
Time (s)
0
1
2
3
Experiment #1
A2
10°
5°
4°
20°
15°
10°
5°
0°
20°
15°
10°
5°
0°
2°
0°
0°
Input amplitude A
2
Output amplitude A
40
Continuum model
Discrete model
Experiments
a
Experiment #2
c
fg
d
Stretching
Shearing
Bending
2 cm
Transmission A40 / A2Cor [2 (t), 40 (t)]
Impactor
#1
#2
#1
#2 #1
la
ui
ui + 1
kl
ks
k
lh
a
A40
Fig. 1 Propagation of elastic vector solitons in a chain with all horizontal hinges aligned. aFew units of our sample (Scale bar: 2 cm). bSchematics of our
testing setup. cEvolution of the rotation and longitudinal displacement of the second and fortieth units as a function of time during two different
experiments. dSchematic of the system. eMeasured transmission, A
40
/A
2
, as a function of the amplitude of the input signal, A
2
.fMeasured cross-
correlation of θ
2
(t) and θ
40
(t) as a function of the amplitude of the input signal, A
2
.gEvolution of the pulse velocity cas a function of its amplitude. The
gray region in e–ghighlights the amplitude gap as predicted by the continuum model. The error bars in gshow the 95% confident interval of the measured
velocities and amplitudes of solitons in experiments. The corresponding error bars for simulation results are too small to show
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solutions. Such amplitude gap is reported as shaded area in
Fig. 1e, f and is in excellent agreement with our numerical and
experimental results. Note that this gap (and the associated
amplitude threshold A
upper
below which solitary waves cannot
propagate) is fundamentally different from the nonlinear
supratransmission effect (limited to weakly nonlinear periodic
waves with certain frequency19), classical amplitude-dependent
dissipation20 and so-called “sonic vacuum”found in not
precompressed granular chains21. The amplitude gaps for solitons
reported in our work are robust features of the system,
intrinsically determined by its architecture and its ability to
support elastic vector solitons.
Looking into the mechanism behind the emergence of this
amplitude gap, it is important to note that the propagation of
vector solitons requires a strong coupling among different
polarizations22,23. However, Eq. (2) show that the coefficients of
the coupling terms in our structure are proportional to θ, so that
large enough rotations are needed in order to activate them and
enable the propagation of vector solitons (see Supplementary
Note 5). Finally, to further verify the validity of our continuum
model, in Fig. 1g we compare the relation between Aand cas
predicted by our analysis (magenta line) and measured in our
experiments (triangular markers) and numerical direct simula-
tions (circular markers) and find good agreement among all three
sets of data.
Enhanced tunability via symmetry breaking. Equation (6)
indicates that in our system the width of the amplitude gap can be
controlled by changing K
s
,K
θ
, and α(see Supplementary Fig. 14).
More excitingly, the tunability and functionality of the proposed
mechanical metamaterial can be further enhanced by breaking
the symmetry in each rigid cross to alter the coupling strength
between the two polarizational components. In our system this is
achieved by shifting neighboring horizontal hinges by a tan φ
0
in
vertical direction (see Fig. 2a and Supplementary Movie 4).
While for the chain with all horizontal hinges aligned (for which
φ
0
=0°) the energy cost to rotate any unit in clockwise and
counter-clockwise directions is identical, the hinges shifting (i.e.,
φ
0
≠0°) introduces a disparity between the two directions of
rotation. Under compression in longitudinal direction, for all
units of the shifted chain with the left hinge higher than the right
one it is energetically more favorable to rotate in clockwise
direction, while for the ones with a lower left hinge rotations in
counter-clockwise direction are preferred (Fig. 2a and Supple-
mentary Fig. 6). By extending our analytical model to units with
φ
0
≠0° and assuming for each unit the positive direction of
rotation to be the one that is naturally induced by compression,
we find that such disparity introduced by the asymmetry is
reflected in the amplitude gap (see Supplementary Note 5). For
the aligned chain (i.e., for φ
0
=0°) the upper (A
upper
) and lower
(A
lower
) limits of the amplitude gap are identical in magnitude
(i.e., A
lower
=−A
upper
, so that in Fig. 1c, d we only show A
upper
).
By contrast, as a result of the bias introduced by the hinges
shifting, when φ
0
increases, Alower
jj
and A
upper
become larger and
smaller, respectively (see Supplementary Note 5). We also find
that a critical angle φcr
0exists at which A
upper
vanishes. In struc-
tures with φ0>φcr
0all solitons that induce an energetically favor-
able rotation at the i-th pair of crosses can propagate through the
system, regardless of their magnitude. The validity of our
analysis is confirmed by experiments and numerical simulations
conducted on a chain comprising 2 × 50 and 2 × 150 crosses
characterized by φ
0
=5°, respectively. For such system our con-
tinuum model predicts A
upper
=0° and A
lower
=−20.91°. In
agreement with this analytical prediction, the amplitude trans-
mission ratio and signal shape cross-correlation between the
input, θ
2
(t), and the output, θ
40
(t), measured in both experiments
and discrete simulations significantly drop when the input
00.51 00.10.2
–1 0 1
a
d
Velocity c
Continuum model
Amplitude A
Input amplitude A
2
Output amplitude A
40
b20°
–20°
–40°
20°
10°
–10°
–20°
–30°
0°
20°
10°
–10°
–20°
–30°
0°
Aupper
Alower
0° 5° 10°
0°
c
ef
2 cm
Transmission A40 / A2
Discrete model
Experiments
a
0
0
cr
0
i
ii+1
i+1
ui+1
ui
Cor [2 (t), 40 (t)]
Fig. 2 Propagation of elastic vector solitons in a chain with vertically shifted neighboring horizontal hinges. aSchematic of the system. Neighboring
horizontal hinges are shifted by atan φ
0
in vertical direction. bEvolution of the amplitude gap as a function of the angle φ
0
.cFew units of our sample
characterized by φ
0
=5° (Scale bar: 2 cm). dMeasured transmission, A
40
/A
2
, as a function of the amplitude of the input signal, A
2
.eMeasured cross-
correlation of θ
2
(t) and θ
40
(t) as a function of the amplitude of the input signal, A
2
.fEvolution of the pulse velocity cas a function of its amplitude. The gray
region in band d–fhighlights the amplitude gap as predicted by the continuum model. The error bars in fshow the 95% confident interval of the measured
velocities and amplitudes of solitons in experiments. The corresponding error bars for simulation results are too small to show
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amplitude falls inside the gap (Fig. 2d, e). Moreover, the
numerical and experimental data also closely match the
amplitude-velocity relation predicted by our continuum model
(Fig. 2f), confirming that the hinges shifting induces an asym-
metric gap.
Functional devices based on amplitude gaps. Having discovered
the existence of amplitude gaps in our system, we next focus on
how such effect can be exploited to design new functional devices
to control mechanical signals, and, therefore, to provide new
opportunities for phononic computing and mechanical logic. In
analogy to beam splitters24, which are widely used in photonics to
split an incident light beam into two or more beams, we start by
taking advantage of amplitude gaps to design a soliton splitter—a
device capable of splitting an incoming elastic vector soliton into a
transmitted and a reflected ones. Remarkably, this can be achieved
by simply introducing a pair of stiffer hinges within an aligned
chain with φ
0
=0°. To demonstrate the concept we take our 2 ×
50 sample with φ
0
=0° and introduce two stiffer hinges (made of
polyester sheets with thickness td
h=0.635 mm) to connect the
24th and the 25th pairs of units (Fig. 3a). We find that, if
the amplitude of the input signal is large enough to be outside the
amplitude gap, the excited solitary wave is split into two pulses by
the pair of stiffer hinges (see Fig. 3b). To better understand the
nature of the transmitted and reflected waves, we integrate Eq. (1)
to simulate the response of a chain comprising 1000 units with
φ
0
=0 and a pair of stiffer hinges (with stiffness Kd
sand Kd
θ)
connecting the 500th and 501st rigid crosses. The numerical
results for a chain with Kd
s=Ks=Kd
θ=Kθ=30 are shown in Fig. 3c,
d. We find that the pair of stiffer hinges split the incoming soliton
into two pulses that propagate with stable shape and constant
velocity and that no trains of solitons are generated. As for the
radiation of linear waves, we find that the interaction between the
incoming solitary wave and the stiffer pair of hinges generates
only translational vibrations, since the frequency content of our
solitons overlaps with a low-frequency band gap for rotation.
However, we estimate that only 4% of the energy carried by the
incoming soliton is finally transferred to translational linear
vibrations (see Supplementary Notes 8–10). As such, these results
clearly indicate that our simple structure acts as a splitter for
solitons. Note that this behavior is remarkably different from that
previously observed in unloaded granular chains, where hetero-
geneities have been found to split the propagating solitary wave
into trains of solitons and to generate stress oscillations localized
near the impurities25. Such difference is due to the presence of the
amplitude gap, which in our mechanical metamaterial prevents
fragmentation of the propagating pulse by suppressing the pro-
pagation of small amplitude solitons. To demonstrate this
important point, we simulate the response of a 2 × 1000 chain with
φ
0
=5° and a pair of stiffer hinges in the middle. Note that this
structures enables the propagation of solitary waves of any
amplitude if they induce energetically favorable rotations (since
A
upper
=0° and there is no amplitude gap for such waves). We
find that, when such waves are excited and hit the pair of
stiffer hinges, trains of pulses are generated (see Fig. 3e, f), con-
firming the important role played by the amplitude gap. Finally,
our numerical results also indicate that our soliton splitter is a
robust device, since the ratio between the energy carried by the
transmitted and reflected solitons only depends on the ratio
Kd
s=Ks=Kd
θ=Kθand not on the amplitude of the input signal, with
the amount of reflected energy monotonically increasing with the
stiffness ratio (see Supplementary Fig. 23).
4
6
8
Normalized time TNormalized time T
01000500
6
8
10
T = 8400
d
0 100 200 300 400 500 600 700 800 900 1000
Unit number i
T = 5600
T = 4800
T = 9200
T = 7200
T = 6000
2515
10
50
100
Time t (ms)
150
ab
35
20 30
Unit number i
Unit number i
1
01000
500 0 100 200 300 400 500 600 700 800 900 1000
Unit number i
Unit number i
f
c
e
2 23242526 4950
Unit number i
K, KsK, Ks
d
d
8°
4°
0°
12°
Angle i
Angle i
Angle i
8°
4°
0°
12°
10°
10°
0°
0°
20°
20°
Angle i
×103
×103
0 = 0°
0 = 0°
0 = 5°
0 = 5°
Fig. 3 Soliton splitter. aSchematics of our soliton splitter. A pair of stiffer hinges (with stiffness Kd
sand Kd
θ) is introduced to connect the 24th and the 25th
pairs of crosses. bRotation of the pairs of crosses during the propagation of the pulse, as recorded with our high-speed camera. The location of the stiff pair
of hinges is indicated by the dashed red line. c,dSimulations corresponding to the experiments shown in b. The numerical analysis are conducted on a 2 ×
1000 chain with symmetric crosses characterized by φ
0
=0° and a pair of stiffer hinges placed between the 500th and the 501st units. e,fNumerical
results for a 2 × 1000 chain with asymmetric crosses characterized by φ
0
=5° and a pair of stiffer hinges placed between the 500th and the 501st units
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Further, we design a mechanical diode26 for solitary waves—a
system that is transparent to solitons incoming from one
direction but blocks those propagating in the other one. While
such nonreciprocal wave transmission has been previously
reported for periodic waves27–29, irreversible transition waves30
and wave packets31, here we extend the concept to solitary wave
pulses. To achieve this, we introduce a few pairs of crosses with
φ
0
≠0 within a chain with φ
0
=0°. More specifically, our diode
comprises two external sections with 2Nand (2N+1) pairs of
crosses characterized by φ
0
=0 and a central portion consisting of
2N
a
pairs of crosses with φ
0
=5° (Fig. 4a). Experiments
conducted on a sample with N=12, N
a
=3 and the section with
2N+1 units placed on the left show that a pulse initiated at the
left end propagates through the entire structure (Fig. 4b, c), while
a solitary wave excited at the right end is completely reflected by
the boundary between the regions with φ
0
=0° and 5° (Fig. 4d, e).
This remarkable behavior is induced by the asymmetric
amplitude gap of the region with φ
0
=5°. Solitons excited at
the left end of the chain induce energetically favorable rotations at
the units with φ
0
=5° and, since there is no amplitude gap for
such waves (i.e., Aφ0¼5
upper =0), they are able to propagate through
the entire chain (Fig. 4f). In contrast, solitons initiated at the right
end of the chain result in energetically unfavorable rotations for
the crosses with φ
0
=5° and, since Aφ0¼5
lower Aφ0¼0
lower , they are
almost completely blocked by the boundary between aligned and
shifted crosses (Fig. 4f). To explore the performance of our
50
100
150
0° 5° 10°
Angle ⎢i⎢
Transmission
⎢A40⎪/ ⎢A10⎪
Transmission
⎢A10⎪/ ⎢A40⎪
Input amplitude ⎢A10⎪
Time t (ms)
Time t (ms)
Amplitude A
100
120
140
160
d
a
10
10°
122N2N + 1 2N + 2Na + 1 4N + 2Na + 1
15° 20° 25°
Input amplitude ⎢A40⎪
10° 15° 20° 25°
20 30 40
Unit number i
b c
0
0.2
0.4
0.6
0.8
Abr
1
0
0.2
0.4
0.6
0.8
20°
–20°
0°
1
t = 58 ms
t = 96 ms
t = 135 ms
t = 158 ms
e
fg
h
Unit number i
Discrete model
Experiments
#1
#2
#1
#2
#1
#2
2 × N + 1 2 × Na2 × N
Fig. 4 Mechanical diode. aSchematics of our mechanical diode. bOptical images showing the propagation of a solitary wave excited at the left end of the
chain. cRotation of the pairs of crosses induced by a pulse excited at the left of the chain. dOptical images showing the propagation of a solitary wave
excited at the right end of the chain. eRotation of the pairs of crosses induced by a pulse excited at the right of the chain. fSchematic highlighting the
working principles of our mechanical diode. gMeasured transmission, A40=A10
jj
, as a function of the input amplitude, A10
jj
, for pulses excited at the left
end of the chain. hMeasured transmission, A10=A40
jj
, as a function of the input amplitude, A40
jj
, for pulses excited at the right end of the chain
ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-05908-9
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mechanical diode, we apply different input signals at the left and
right end of the chain in both experiments and discrete
simulations. We find that for all pulses with amplitude larger
than Aφ0¼0
upper ¼6:55initiated at the left end of the system the
transmission, A40
jj=A10
jj, approaches unity (Fig. 4g). Differently,
when the excitation is applied at the right end of the chain, the
transmission, A10
jj=A40
jj, is close to zero even if the amplitude of
the input signal is outside the gap of the region with φ
0
=0 (i.e.,
A40
jj>Aφ0¼0
upper). However, as typically observed in electronic and
thermal diodes32, if the amplitude of the pulses becomes too large,
the diode experiences a condition known as breakdown. As a
result, solitary waves with amplitude larger than A
br
≈15°
propagate through the diode (i.e., if A40
jj>Abr ≈15°, then
A10
jj
=A40
jj
~ 0.6—see Supplementary Note 11 for a detailed
numerical study on the dependency of A
br
on φ
0
and N
a
).
Discussion
In this study, we have experimentally observed, numerically
simulated, and mathematically analyzed the existence of ampli-
tude gaps for elastic vector solitons in highly deformable
mechanical metamaterials consisting of rigid units and elastic
hinges. First, we have shown that such amplitude gaps can be
tuned by altering both the structural parameters and the sym-
metry of the crosses. Then, we have demonstrated that amplitude
gaps can be exploited to design clean splitters and diodes for
highly nonlinear solitary waves. In recent years, many strategies
have been proposed to manipulate the propagation of elastic
waves29,33, enabling a wide range of applications such as spatial
guiding34–36, frequency filtering37–39, noise/impact mitiga-
tion40,41, and non-reciprocal transmission14,42. However, the vast
majority of devices focus on small-amplitude vibrations and take
advantage of spectral gaps in frequency29,33,43. As such, our study
on amplitude gaps for highly nonlinear solitary waves adds a
whole new dimension to our ability to design structures and
materials with tailored dynamic behavior and open avenues for
potential technological breakthroughs.
Methods
Summary of Supplementary Note. Details on fabrication are provided in Sup-
plementary Note 1; on experiments in Supplementary Note 2; on the discrete
model in Supplementary Note 3; on the continuum model in Supplementary
Note 4; on amplitude gaps for solitons in Supplementary Note 5; on the solution
for the aligned chain in Supplementary Note 6; on solitons excited by pulling in
Supplementary Note 7; on energy carried by solitons in Supplementary Note 8 and
on the dispersion relation in Supplementary Note 9. Furthermore, additional
results for the splitter and the diode are shown in Supplementary Notes 10 and 11,
respectively.
Code availability. Matlab code for numerical simulations are included in
the Supplementary Files.
Data availability. The authors declare that data supporting the findings of this
study are included within the paper and its Supplementary Information files or are
available from the corresponding author upon reasonable request
Received: 28 June 2018 Accepted: 25 July 2018
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Acknowledgements
K.B. acknowledges support from the National Science Foundation under Grant No.
DMR-1420570 and EFMA-1741685 and from the Army Research Office under Grant
No. W911NF-17-1-0147.
Author contributions
B.D., P.W., V.T. and K.B. conceived the project. The experiments were conducted by
B.D., Q.H. and V.T. B.D. wrote the code for numerical simulation in MATLAB. All the
authors contributed to the derivation and completion of the amplitude gap theory. B.D.,
P.W., V.T. and K.B. wrote the manuscript with input from all authors.
Additional information
Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467-
018-05908-9.
Competing interests: The authors declare no competing interests.
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