# 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.

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**ABSTRACT:**We propose a type of locally resonant structure involving arrays of structured coated inclusions. The coating consists of a structural interface with beams inclined at a certain angle. Such an elastic metamaterial supports tunable low-frequency stop bands associated with localized rotational modes that can be used in the design of filtering, reflecting, and focusing devices. Asymptotic estimates for resonant frequencies are in good agreement with finite element computations and can be used as a design tool to tune stop band changing relative inclinations, number, and cross section of the beams. Inertial resonators with inclined ligaments allow for anomalous dispersion (negative group velocity) to occur in the pressure acoustic band and this leads to the physics of negative refraction, whereby a point force located above a finite array of resonators is imaged underneath for a given polarization. We finally observe that for a periodic macrocell of the former inertial resonators with one defect in the middle, an elastic trapped mode exists within a high-frequency stop band. The latter design could be used in the enhancement of light and sound interactions in photonic crystal fiber preforms.Physical review. B, Condensed matter 05/2013; 87:174303. · 3.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Here in locally resonant acoustic material, it is shown that effective zero refractive index can be constructed by the resonant unit-cells with coherent degenerate monopole–dipole momenta. Due to strong local resonances, the material layers with effective zero refractive index can function as a resonant cavity of high Q factor, where a subtle deviation from the resonant frequency may result in distinct increase of reflection. Full-wave simulations are performed to demonstrate some unusual wave transport properties such as invisibility cloaking, super-reflection, local field enhancement, and wavefronts rotation.Physics Letters A 10/2013; 377(s 31–33):1784–1787. · 1.77 Impact Factor - SourceAvailable from: Andrea Bergamini[Show abstract] [Hide abstract]

**ABSTRACT:**The band structure of a phononic crystal can be controlled by tuning the mechanical stiffness of the links connecting its constituting elements. The first implementation of a phononic crystal with adaptive connectivity is obtained by using piezoelectric resonators as variable stiffness elements, and its wave-propagation properties are experimentally characterized.Advanced Materials 03/2014; 26(9):1343-7. · 14.83 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 Zhao-Qing 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 double-negative 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

withthedouble-negativeelectromagneticmaterials.

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 ‘fluid-like’. 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 slow-transverse phonons can create

abandgapinsidetheBrillouinzoneinadditiontothoseatthezone

boundary and the zone centre27.

ouble-negative 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 23955-6900, Kingdom of Saudi Arabia. *e-mail:phzzhang@ust.hk.

Foam

host

Steel

m

m

m

m

Hard silicone

rubber

Soft silicone

rubber

ab

Figure 1|Physical model and a practical design. a, The physical model of a

type of multi-mass 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 two-dimensional elastic metamaterial with

its unit cell comprising a multi-mass 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 double-negative 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 ‘fluid-like’ and

‘super-anisotropic’, 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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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 2|Band 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

low-frequency 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 high-frequency 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 3|Field 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 single-mass 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 4|Transmission through a finite sample. a, The numerical set-up 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 non-zero 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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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 ‘fluid-like’. 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

‘fluid-like’ in certain directions and ‘incompressible-solid-like’

(that is, allowing only shear waves) in certain other directions, a

property which is denoted ‘super-anisotropic’. In the following, we

performtransmissioncalculationsbyusingCOMSOLMultiphysics

to illustrate the transmission properties of the hybrid bands. A

schematicforthenumericalset-upinCOMSOLisshowninFig. 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 ‘super-anisotropic’ 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 ‘super-anisotropic’ 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 finite-size 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 ‘fluid-like’ and ‘super-anisotropic’ 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 double-negative 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 Z-Q.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 Z-Q.Z.

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