Hybrid elastic solids.
ABSTRACT Metamaterials can exhibit electromagnetic and elastic characteristics beyond those found in nature. In this work, we present a design of elastic metamaterial that exhibits multiple resonances in its building blocks. Band structure calculations show two negative dispersion bands, of which one supports only compressional waves and thereby blurs the distinction between a fluid and a solid over a finite frequency regime, whereas the other displays 'super anisotropy' in which compressional waves and shear waves can propagate only along different directions. Such unusual characteristics, well explained by the effective medium theory, have no comparable analogue in conventional solids and may lead to novel applications.

Article: Negative refraction of elastic waves at the deepsubwavelength scale in a singlephase metamaterial
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ABSTRACT: Supplementary information available for this article at http://www.nature.com/ncomms/2014/141124/ncomms6510/suppinfo/ncomms6510_S1.htmlNature Communications 11/2014; 5. · 10.74 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Directing acoustic waves along curved paths is critical for applications such as ultrasound imaging, surgery and acoustic cloaking. Metamaterials can direct waves by spatially varying the material properties through which the wave propagates. However, this approach is not always feasible, particularly for acoustic applications. Here we demonstrate the generation of acoustic bottle beams in homogeneous space without using metamaterials. Instead, the sound energy flows through a threedimensional curved shell in air leaving a closetozero pressure region in the middle, exhibiting the capability of circumventing obstacles. By designing the initial phase, we develop a general recipe for creating selfbending wave packets, which can set acoustic beams propagating along arbitrary prescribed convex trajectories. The measured acoustic pulling force experienced by a rigid ball placed inside such a beam confirms the pressure field of the bottle. The demonstrated acoustic bottle and selfbending beams have potential applications in medical ultrasound imaging, therapeutic ultrasound, as well as acoustic levitations and isolations.Nature Communications 07/2014; 5:4316. · 10.74 Impact Factor  SourceAvailable from: MingHui LuXu Ni, Ying Wu, ZeGuo Chen, LiYang Zheng, YeLong Xu, Priyanka Nayar, XiaoPing Liu, MingHui Lu, YanFeng Chen[Show abstract] [Hide abstract]
ABSTRACT: We numerically realize the acoustic rainbow trapping effect by tapping an air waveguide with spacecoiling metamaterials. Due to the high refractiveindex of the spacecoiling metamaterials, our device is more compact compared to the reported trappedrainbow devices. A numerical model utilizing effective parameters is also calculated, whose results are consistent well with the direct numerical simulation of spacecoiling structure. Moreover, such device with the capability of dropping different frequency components of a broadband incident temporal acoustic signal into different channels can function as an acoustic wavelength division demultiplexer. These results may have potential applications in acoustic device design such as an acoustic filter and an artificial cochlea.Scientific Reports 11/2014; 4:7038. · 5.08 Impact Factor
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ARTICLES
PUBLISHED ONLINE:26 JUNE 2011  DOI:10.1038/NMAT3043
Hybrid elastic solids
Yun Lai1,2, Ying Wu1,3, Ping Sheng1and ZhaoQing Zhang1*
Metamaterials can exhibit electromagnetic and elastic characteristics beyond those found in nature. In this work, we present
a design of elastic metamaterial that exhibits multiple resonances in its building blocks. Band structure calculations show two
negative dispersion bands, of which one supports only compressional waves and thereby blurs the distinction between a fluid
and a solid over a finite frequency regime, whereas the other displays ‘super anisotropy’ in which compressional waves and
shearwavescanpropagateonlyalongdifferentdirections.Suchunusualcharacteristics,wellexplainedbytheeffectivemedium
theory, have no comparable analogue in conventional solids and may lead to novel applications.
D
within a certain frequency regime. The initial proposal1and its
more recent realizations2–8permit the index of refraction to take
negative values, with broad scientific and practical implications9–16.
The ability of doublenegative metamaterials to possess unusual
electromagnetic responses is bestowed by the special resonances
provided by its unit structure. This principle is equally valid for
acousticandelasticmaterials.Anacousticmetamaterialcomprising
locally resonant units17was shown to exhibit negative mass
density18,19; and negative bulk modulus was demonstrated in a
system of Helmholtz resonators20. By combining the two, various
schemes have been proposed and implemented to realize a double
negative characteristic for the compressional wave21–25, in analogy
withthedoublenegativeelectromagneticmaterials.
The ability to withstand shear is a trait that distinguishes a solid
from a liquid. In an elastic solid, the increased number of relevant
material parameters, when combined with the possibility of double
negativity, can yield characteristics that are much more complex
than those seen in electromagnetic and acoustic metamaterials,
some of which, as we show here, can blur the delineating feature
betweenasolidandaliquidwithinacertainfrequencyregime.
For a solid in a periodic structure, the dispersion is in general
anisotropic. Even in the simplest case of a square lattice, one must
take into consideration the realization of negative values for not
only mass density ρ, but also three elastic moduli26, namely c11,
c12 and c44, as well as the various possible interactions between
these parameters. Intriguing possibilities arise. For example, if
both mass density and compressional wave moduli are negative
within a certain frequency regime, then one may have only a
negative band propagating compressional waves (and evanescent
shear waves), which makes a solid ‘fluidlike’. Another possibility is
to have negative dispersions for the compressional wave and shear
wave along distinct directions. The potential realization of such
possibilities, or even a subset, would open new horizons in solid
wave mechanics. Here we would like to mention that anisotropy in
semiconductor superlattices can also give rise to some interesting
phenomenon. It was shown that hybridization of longitudinal
acoustic phonons and folded slowtransverse phonons can create
abandgapinsidetheBrillouinzoneinadditiontothoseatthezone
boundary and the zone centre27.
oublenegative electromagnetic materials denote those
artificial structures in which both the dielectric constant ε
and magnetic permeability µ are simultaneously negative
1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China,2Department of Physics,
Soochow University, 1 Shizi Street, Suzhou 215006, China,3Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of
Science and Technology, Thuwal 239556900, Kingdom of Saudi Arabia. *email:phzzhang@ust.hk.
Foam
host
Steel
m
m
m
m
Hard silicone
rubber
Soft silicone
rubber
ab
Figure 1Physical model and a practical design. a, The physical model of a
type of multimass resonating unit cell that can support monopolar, dipolar
and quadrupolar resonances, and lead to novel elastic properties. b, A
realistic elastic metamaterial unit that is designed according to the physical
model in a.
Aunitcellwithmultipleresonances
We propose a type of twodimensional elastic metamaterial with
its unit cell comprising a multimass locally resonant inclusion that
can generate resonances with monopolar, dipolar and quadrupolar
characteristics. The proposed unit cell is shown to lead to negative
values, not only for mass density, but also for certain elastic
moduli. With physically realizable material parameters, we use
finite element simulations to demonstrate that when these unit
cells are arranged in a simple square lattice, there can be two
hybridized bands with novel characteristics. One of the hybridized
band lies in the doublenegative frequency regime for mass density
and compressional wave moduli, so that only longitudinal pressure
waves can propagate (with negative dispersion), whereas transverse
shear waves decay exponentially. In the other band it is found
that along distinct directions only longitudinal pressure waves
or transverse waves are allowed, both with negative dispersions.
These phenomena, absent in nature, are denoted as ‘fluidlike’ and
‘superanisotropic’, respectively.
A schematic figure of the physical model of the unit cell is
shown in Fig. 1a. The model is a mass–spring system composed
of four masses connected to each other and to the host by
springs. Collective motion of the four masses can enhance the
dipolar resonance (for negative mass density), whereas their
relative motions can enhance the quadrupolar and monopolar
resonances (for negative moduli). A practical realization of the
model in Fig. 1a is illustrated in Fig. 1b. The resonant inclusions
620
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NATURE MATERIALS DOI:10.1038/NMAT3043
ARTICLES
B
A
0
50
100
150
200
250
300
ΜΓΧ
f (Hz)
f (Hz)
f (Hz)
eff (N m¬2)
ΓΧ
ΓΜ
ΓΧ
ΓΜ
204 206208 210 212 214
¬1 × 109
¬2 × 109
¬2 × 109
¬4 × 109
¬6 × 109
0
168 170 172 174 176
0
eff (N m¬2)
μ
a
b
c
κ
Figure 2Band structure and effective medium parameters. a, The band
structure of the elastic metamaterial. Two distinct negative bands are
marked by red and blue dots, respectively. The crosses indicate the
dispersions obtained by using the effective medium theory. b, The κefffor
the higher frequency negative band, exhibiting large negative values at the
upper band edge, that is, close to the ? point. c, The µefffor the lower
frequency negative band, exhibiting large negative values at the upper band
edge, that is, close to the ? point.
are composed of a soft silicone rubber rod embedded with a
hard silicone rubber rod, surrounded by four rectangular steel
rods. The matrix material is chosen to be a foam that has a
light mass density as well as low moduli. The square lattice
has a lattice constant of 10cm; the radii of the soft and hard
siliconerubberrodsare4cmand1cm,respectively;therectangular
steel rods are 1.6cm × 2.4cm in size, located at a distance of
2.4cm from the centre. The material parameters are taken to be
ρ =115kgm−3, λ=6×106Nm−2, µ=3×106Nm−2for foam28;
ρ =1.3×103kgm−3, λ=6×105Nm−2, µ=4×104Nm−2for soft
silicone rubber17; ρ = 1.415×103kgm−3, λ = 1.27×109Nm−2,
µ = 1.78×106Nm−2for hard silicone rubber (J. Page, personal
communication); ρ = 7.9 × 103kgm−3, λ = 1.11 × 1011Nm−2,
µ = 8.28×1010Nm−2for steel. At certain frequencies, the four
rectangular steel rods serve as the four masses, while the silicone
rubber rods serve as the springs in Fig. 1a. The insertion of the
hard silicone rubber is for the purpose of adjusting the spring
constants between the masses.
Bandstructureandeigenstates
ThebandstructureoftheelasticcompositeinFig. 1bwascalculated
by using a finite element solver (COMSOL Multiphysics) and is
shown in Fig. 2a. There are two bands (red and blue dots) with
negative curvatures. The lower frequency negative dispersion band
(red dots) has a bandwidth of about 18Hz. A small complete
gap (178Hz–198Hz) separates it from a higher frequency negative
dispersion band (blue dots) which has a bandwidth of about
18Hz along the ?M direction but only 10Hz in the ?X direction.
There is also a complete gap above the higher negative band
(216Hz–255Hz). We note that, around 200Hz, the transverse and
longitudinal wavelengths in the foam host are, respectively, about
80cm and 160cm; much larger than the lattice constant of 10cm.
Thus, these negative bands are not induced by Bragg scattering but
are rather the results of hybridization between different types of
resonances within the unit cell.
An investigation of the eigenstates in the bands gives us a clear
picture of the physical origin of the bands. The eigenstates in the
lowfrequency bands, delineated in Fig. 2a by dark yellow, red and
bluedots,havetheirkineticenergy(bothvibrationalandrotational)
mostly concentrated in the steel rods. In contrast, for eigenstates in
the highfrequency bands, delineated in Fig. 2a by black dots, the
energy is mostly concentrated inside the soft silicone rubber. The
origin of the band gaps shown in Fig. 2a is the collective motions
+
+
a
c
e
g
b
d
f
h
Figure 3Field distributions of some specific eigenstates. a,b, The
eigenstates (f =178.5Hz and 216.8Hz) at the ? point in the lower and
upper negative bands, respectively. Here, arrows indicate displacements
and colour indicates amplitude (red for large and blue for small). c,d, The
displacements of the quadrupolar and monopolar resonances that
correspond to a and b, respectively. e, The eigenstate (f =169.4Hz) at the
midpoint between the ? and M points in the lower negative band (marked
by A in Fig. 2a). f, The eigenstate (f =210.3Hz) at the midpoint between
the ? and X points in the upper negative band (marked by B in Fig. 2a).
g,h, The states in e and f are shown to arise as hybridizations,
(quadrupole+dipole) and (monopole+dipole), respectively.
of steel rods that enhance dipolar resonances and thus produces a
negativemassdensitywithinthefrequencyrangeof160Hz–255Hz,
in a manner similar to singlemass resonator metamaterials17.
However, as well as the dipolar resonance, the relative motions of
steel rods can support two other important resonances, namely,
the monopolar and quadrupolar resonances, which are found to be
responsible for the two negative bands (blue and red dots) inside
the range of negative mass density. In Fig. 3a and b, we plot the two
eigenstates in the lower and upper negative bands (f = 178.5Hz
and 216.8Hz) at the ? point, respectively. Here, the colour
represents the amplitude of displacement (blue/red for small/large
values) and the arrows show the displacement vectors directly. The
eigenstate in Fig. 3a is clearly a quadrupolar resonance, whereas the
eigenstate in Fig. 3b is a monopolar resonance. Schematics of the
two resonances are shown by the blue thick arrows in Fig. 3c and
d, respectively, which indicate the displacements in the positions
of the steel rods and exhibit clear quadrupolar and monopolar
signatures. Thus, we can view the two negative bands as being
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NATURE MATERIALS DOI:10.1038/NMAT3043
PML
Data line
Force
100 120140160180 200 220 240 260 280
f (Hz)
100120140160 180200 220 240 260 280
f (Hz)
100 120 140160180200 220 240 260 280
f (Hz)
100 120140160180 200 220 240 260 280
f (Hz)
10¬7
10¬6
10¬5
10¬4
10¬3
10¬2
10¬1
100
101
10¬7
10¬6
10¬5
10¬4
10¬3
10¬2
10¬1
100
101
10¬5
10¬4
10¬3
10¬2
10¬1
100
101
10¬8
10¬7
10¬6
10¬5
10¬4
10¬3
10¬2
10¬1
100
101
Transmission (ΓΜ)
Transmission (ΓΜ)
Transmission (ΓΧ)
Transmission (ΓΧ)
Transverse input (
p
s )
Transverse input (
p
s )
Longitudinal input (
p
s )
Longitudinal input (
p
s )
a
b
d
c
e
Figure 4Transmission through a finite sample. a, The numerical setup for transmission computations. b,c, Transmission along the ?M direction for
transverse input excitations (b) and longitudinal input excitations (c). Transmission along the ?X direction for transverse input excitations (d) and
longitudinal input excitations (e). In the upper frequency negative dispersion band, indicated by the blue dashed lines, it is seen that whereas longitudinal
excitations can lead to large transmissions along both the ?M and ?X directions (c,e), strong attenuation is seen for transverse input excitations (b,d). For
the lower frequency negative dispersion band, indicated by the red dashed lines, large transmissions are seen in b and e, in exact agreement with the
effective medium predictions.
inducedbyquadrupolarandmonopolarresonancesinsidetheband
gap created by the dipolar resonance. This may be viewed as the
hybridization effect of the quadrupolar/monopolar resonance with
the dipolar resonance.
To directly see the hybridization effect, we plot in Fig. 3e and
f the state at the midpoint between the ? and M points in the
lower negative band (marked by the symbol ‘A’ with f =169.4Hz)
and the state at the midpoint between the ? and X points in the
upper negativeband (marked by thesymbol ‘B’ withf =210.3Hz),
respectively. It is seen that away from the ? point, the pure
quadrupolarormonopolarstatesturnintohybridstatesthatcanbe
regarded as combinations of a monopolar or quadrupolar state and
a dipolar state. These are illustrated in Fig. 3g and h, respectively.
These hybrid states have negative nonzero group velocities and
thus can transmit energy.
Effective medium description
The formation of negative bands can be understood from an
effective medium point of view. It is known that dipolar reso
nances can lead to negative mass density17–19, whereas monopolar
and quadrupolar resonances are associated with certain elastic
moduli28,29. From the theory of linear elasticity, the three elastic
moduliforasolidwithasquarelatticearec11,c12andc44.Thedisper
sions and the associated modes can be obtained from the Christof
fel’s equation26. Along the ?X direction, compressional wave and
shear wave velocities are given by√c11/ρ and√c44/ρ, respectively;
whereas along the ?M direction the compressional and shear wave
velocities are√(c11+c12+2c44)/(2ρ) and√(c11−c12)/(2ρ), re
spectively. Thus, we can predict the transport properties of this hy
brid elastic solid if we can obtain the effective medium parameters.
Owingtothestronglyanisotropicnatureoftherelativemotionsthat
are possible within our unit cell, an effective medium theory that
relies on the surface motion/response to external stimuli (in con
trasttovolumeaverage),isshowntobemoregenerallyapplicableto
calculatetheeffectiveparametersalongboth?Xand?Mdirections.
Alongthe?Xand?Mdirections,wefindthatitisconvenientto
introduceκeff=(ceff
elastic bulk modulus and shear modulus that correspond with
the monopolar and quadrupolar resonances, respectively. Whereas
ρeff= −Feff
unit cell and its effective displacement ueff
surface integration of the stresses (along the x direction) and the
displacements over the unit cell, the effective moduli are evaluated
from the effective stress and strain relations: Teff
Teff
stresses and the effective strains are evaluated on the unit cell
boundary.DetailscanbefoundintheSupplementaryInformation.
In the lower negative band, the µeffevaluated from the relevant
eigenstates is negative and diverges (in the negative direction) at
the ? point owing to the quadrupolar resonance, whereas κeffis
11+ceff
12)/2andµeff=(ceff
11−ceff
12)/2astheeffective
x/ω2ueff
xa2, where both the effective force Feff
x
on the
xmay be obtained from
xx=ceff
11Seff
xx+ceff
12Seff
yy,
yy= ceff
12Seff
xx+ceff
11Seff
yy, and Teff
xy= 2ceff
44Seff
xy, where both the effective
622
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© 2011 Macmillan Publishers Limited. All rights reserved
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NATURE MATERIALS DOI:10.1038/NMAT3043
ARTICLES
positive and finite. In the higher negative band, κeffis negative
and diverges (in the negative direction) at the ? point owing to
the monopolar resonance, whereas µeffis positive and finite. The
compressional wave and shear wave velocities along the ?X (?M)
direction can be obtained by√(κeff+µeff)/ρeff(√(κeff+ceff
and
can be clearly seen that in the higher frequency negative dispersion
band, the fact that κeff? 0, ρeff< 0, µeff> 0 and ceff
only the longitudinal wave can propagate along the ?X and ?M
directions. For the lower frequency negative dispersion band we
have µeff? 0, ρeff< 0 and other parameters positive, therefore
onlythelongitudinal(transverse)wavecanpropagatealongthe?X
(?M) direction. The resulting effective medium predictions of the
dispersion relations are shown as crosses in Fig. 2a. The agreement
is excellent. It should be noted, however, that owing to the nature
of quadrupolar excitation, ceff
be obtained from the effective medium theory. Thus the condition
ceff
44)/ρeff)
√ceff
44/ρeff(√µeff/ρeff), respectively. From these formulas, it
44> 0 implies
44for the lower negative band cannot
44>0isinferredbyconsistencywiththebandstructures.
Discussionofnovelcharacteristics
The above effective medium analysis predicts that this solid
metamaterial has very unusual acoustic properties, beyond those of
normal solids. In the higher frequency negative dispersion band,
the elastic metamaterial can only transport pressure waves and
thus turns ‘fluidlike’. Therefore, its impedance can be perfectly
matched to a fluid host or soft tissues where pressure waves
dominate. In the lower frequency negative dispersion band, the
metamaterial turns into a very unique anisotropic solid that is
‘fluidlike’ in certain directions and ‘incompressiblesolidlike’
(that is, allowing only shear waves) in certain other directions, a
property which is denoted ‘superanisotropic’. In the following, we
performtransmissioncalculationsbyusingCOMSOLMultiphysics
to illustrate the transmission properties of the hybrid bands. A
schematicforthenumericalsetupinCOMSOLisshowninFig. 4a.
Foraslabofmetamaterialwithsevenlayersalongthe?Mdirection,
an external normal/tangential harmonic force is exerted on the
left side to provide an input of longitudinal/transverse waves. A
perfect matched layer (PML) is added at the right side and periodic
boundary conditions are imposed on the upper and lower edges.
Thetransmissionsforthelongitudinalortransversewaves(denoted
as ‘p’ or ‘s’) can be calculated by integrating the horizontal or
vertical displacements on the data line. In Fig. 4b,c, we show the
obtained transmissions in the ?M direction under transverse and
longitudinal inputs, respectively. It is seen that large transmissions
(on the order of one) for p waves are obtained for the upper
negative dispersion band (delineated by blue dashed lines) under
a longitudinal input. By contrast, large transmissions for s waves
are obtained for the lower negative dispersion band (delineated by
red dashed lines) under a transverse input. In Fig. 4d,e, we show
theobtainedtransmissionsalongthe?Xdirectionundertransverse
and longitudinal inputs, respectively. Large transmissions are seen
onlyforlongitudinalinputexcitations,forbothnegativedispersion
bands. The results are in exact agreement with the predictions of
the effective medium theory. Here we note that there are some
transmission values greater than one in Fig. 4b–e caused by the
use of force (load) as input instead of incident waves, but these
(normalizationissues)donotaffecttheanalysishere.
The ‘superanisotropic’ behaviour can be understood as a result
of symmetry breaking in the rubber rod due to the presence of four
steel rods. As shown in Fig. 3c, a local quadrupolar resonance in a
cylindrically symmetric rubber rod (without the steel rods) has two
degeneratemodesatthe? point(withdisplacementsmarkedasred
and blue thick arrows). By matching with the displacements of the
host medium (red and blue thin arrows), it is seen that one mode
(red arrows) is transverse in the ?X direction and longitudinal in
the ?M direction, whereas the other mode (blue arrows) is just the
opposite. When the steel rods are inserted at the positions of the
dashedboxes,thesymmetryoftherubberrodisbrokenandthetwo
degenerate states split. The one denoted by blue arrows in Fig. 3c
leads to the lower negative band with the ‘superanisotropic’ elastic
characteristic shown in Fig. 2a.
We note that, in general, the theory of linear elasticity needs to
be modified so as to accommodate the finitesize limitation of the
unit cell in elastic metamaterials30–32. During the effective medium
calculation of the lower negative band, we have actually observed
stresses Teff
elastic theory. However, this ‘local rotation’ effect does not have
a major impact on the phenomena shown in this paper, as they
are actually determined by the resonances involving the diagonal
terms, namely Teff
κeffand µeffin Fig. 2b and c, respectively. Nevertheless, the ‘local
rotation’ effect may have some interesting implications in other
types of metamaterials30–32.
The ‘fluidlike’ and ‘superanisotropic’ hybrid elastic solids
shown in this report represent two types of new solids absent in
nature.Theysignificantlyextendourabilitytocontrolelasticwaves
insolids.Potentialapplicationsofthesehybridelasticsolidsinclude
wave polarizers, wave imaging and confinement33, controlling
elastic and seismic waves34,35, transformation acoustics36, and so
on. As a result of their doublenegative nature, negative refraction
andsuperlensingforlongitudinalortransversecomponentsarealso
possible. Having a richer variety of unusual properties than their
electromagnetic and acoustic counterparts, elastic metamaterials
are likely to generate further new ideas and novel applications in
the near future.
Received 17 January 2011; accepted 6 May 2011; published online
26 June 2011
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Acknowledgements
We thank Z. Hang and I. Tsukerman for useful discussions. This work was supported by
Hong Kong RGC Grant No. 605008 and RGC Grant HKUST604207.
Author contributions
Y.L. and Y.W. carried out the research and contributed equally. P.S. and ZQ.Z.
supervised the research and contributed to its design. All the authors discussed the
results extensively.
Additional information
The authors declare no competing financial interests. Supplementary information
accompanies this paper on www.nature.com/naturematerials. Reprints and permissions
information is available online at http://www.nature.com/reprints. Correspondence and
requests for materials should be addressed to ZQ.Z.
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