Article
Hybrid elastic solids
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China.
Nature Materials (Impact Factor: 36.5). 06/2011; 10(8):6204. DOI: 10.1038/nmat3043 Source: PubMed
Get notified about updates to this publication Follow publication 
Fulltext
Available from: Pai Peng, Jun 12, 2014ARTICLES
PUBLISHED ONLINE: 26 JUNE 2011  DOI: 10.1038/NMAT3043
Hybrid elastic solids
Yun Lai
1,2
, Ying Wu
1,3
, Ping Sheng
1
and ZhaoQing Zhang
1
*
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.
D
oublenegative electromagnetic materials denote those
artificial structures in which both the dielectric constant ε
and magnetic permeability µ are simultaneously negative
within a certain frequency regime. The initial proposal
1
and its
more recent realizations
2–8
permit the index of refraction to take
negative values, with broad scientific and practical implications
9–16
.
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
acoustic and elastic materials. An acoustic metamaterial comprising
locally resonant units
17
was shown to exhibit negative mass
density
18,19
; and negative bulk modulus was demonstrated in a
system of Helmholtz resonators
20
. By combining the two, various
schemes have been proposed and implemented to realize a double
negative characteristic for the compressional wave
21–25
, in analogy
with the doublenegative electromagnetic materials.
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
between a solid and a liquid within a certain frequency regime.
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 moduli
26
, namely c
11
,
c
12
and c
44
, 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
a band gap inside the Brillouin zone in addition to those at the zone
boundary and the zone centre
27
.
1
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China,
2
Department of Physics,
Soochow University, 1 Shizi Street, Suzhou 215006, China,
3
Division 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 1  Physical 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.
A unit cell with multiple resonances
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 NATURE MATERIALS  VOL 10  AUGUST 2011  www.nature.com/naturematerials
© 2011 Macmillan Publishers Limited. All rights reserved
Page 1
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
ΓΧ
ΓΜ
ΓΜ
204 206 208 210 212 214
¬1 × 10
9
¬2 × 10
9
¬2 × 10
9
¬4 × 10
9
¬6 × 10
9
0
168 170 172 174 176
0
eff
(N m
¬2
)
a
b
c
μ
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 κ
eff
for
the higher frequency negative band, exhibiting large negative values at the
upper band edge, that is, close to the 0 point. c, The µ
eff
for the lower
frequency negative band, exhibiting large negative values at the upper band
edge, that is, close to the 0 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 10 cm; the radii of the soft and hard
silicone rubber rods are 4 cm and 1 cm, respectively; the rectangular
steel rods are 1.6 cm × 2.4 cm in size, located at a distance of
2.4 cm from the centre. The material parameters are taken to be
ρ =115 kg m
−3
, λ = 6 ×10
6
N m
−2
, µ = 3 ×10
6
N m
−2
for foam
28
;
ρ =1.3×10
3
kg m
−3
, λ =6×10
5
N m
−2
, µ =4×10
4
N m
−2
for soft
silicone rubber
17
; ρ = 1.415 × 10
3
kg m
−3
, λ = 1.27 × 10
9
N m
−2
,
µ = 1.78 ×10
6
N m
−2
for hard silicone rubber (J. Page, personal
communication); ρ = 7.9 × 10
3
kg m
−3
, λ = 1.11 × 10
11
N m
−2
,
µ = 8.28 × 10
10
N m
−2
for 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.
Band structure and eigenstates
The band structure of the elastic composite in Fig. 1b was calculated
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 18 Hz. A small complete
gap (178 Hz–198 Hz) separates it from a higher frequency negative
dispersion band (blue dots) which has a bandwidth of about
18 Hz along the 0M direction but only 10 Hz in the 0X direction.
There is also a complete gap above the higher negative band
(216 Hz–255 Hz). We note that, around 200 Hz, the transverse and
longitudinal wavelengths in the foam host are, respectively, about
80 cm and 160 cm; much larger than the lattice constant of 10 cm.
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
blue dots, have their kinetic energy (both vibrational and rotational)
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 3  Field distributions of some speciﬁc eigenstates. a,b, The
eigenstates (f =178.5 Hz and 216.8 Hz) at the 0 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 =169.4 Hz) at the
midpoint between the 0 and M points in the lower negative band (marked
by A in Fig. 2a). f, The eigenstate (f =210.3 Hz) at the midpoint between
the 0 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
negative mass density within the frequency range of 160 Hz–255 Hz,
in a manner similar to singlemass resonator metamaterials
17
.
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 = 178.5 Hz
and 216.8 Hz) at the 0 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
NATURE MATERIALS  VOL 10  AUGUST 2011  www.nature.com/naturematerials 621
© 2011 Macmillan Publishers Limited. All rights reserved
Page 2
ARTICLES
NATURE MATERIALS DOI: 10.1038/NMAT3043
PML
Data line
Force
100 120 140 160 180 200 220 240 260 280
100 120 140 160 180 200 220 240 260 280 100 120 140 160 180 200 220 240 260 280
100 120 140 160 180 200 220 240 260 280
10
¬7
10
¬6
10
¬5
10
¬4
10
¬3
10
¬2
10
¬1
10
0
10
1
10
¬7
10
¬6
10
¬5
10
¬4
10
¬3
10
¬2
10
¬1
10
0
10
1
10
¬5
10
¬4
10
¬3
10
¬2
10
¬1
10
0
10
1
10
¬8
10
¬7
10
¬6
10
¬5
10
¬4
10
¬3
10
¬2
10
¬1
10
0
10
1
f (Hz)
f (Hz)
f (Hz)
f (Hz)
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 setup for transmission computations. b,c, Transmission along the 0M direction for
transverse input excitations (b) and longitudinal input excitations (c). Transmission along the 0X 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 0M and 0X 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.
induced by quadrupolar and monopolar resonances inside the band
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 0 and M points in the
lower negative band (marked by the symbol ‘A’ with f =169.4 Hz)
and the state at the midpoint between the 0 and X points in the
upper negative band (marked by the symbol ‘B’ with f =210.3 Hz),
respectively. It is seen that away from the 0 point, the pure
quadrupolar or monopolar states turn into hybrid states that can be
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 density
17–19
, whereas monopolar
and quadrupolar resonances are associated with certain elastic
moduli
28,29
. From the theory of linear elasticity, the three elastic
moduli for a solid with a square lattice are c
11
, c
12
and c
44
. The disper
sions and the associated modes can be obtained from the Christof
fel’s equation
26
. Along the 0X direction, compressional wave and
shear wave velocities are given by
√
c
11
/ρ and
√
c
44
/ρ, respectively;
whereas along the 0M direction the compressional and shear wave
velocities are
√
(c
11
+c
12
+2c
44
)/(2ρ) and
√
(c
11
−c
12
)/(2ρ), re
spectively. Thus, we can predict the transport properties of this hy
brid elastic solid if we can obtain the effective medium parameters.
Owing to the strongly anisotropic nature of the relative motions that
are possible within our unit cell, an effective medium theory that
relies on the surface motion/response to external stimuli (in con
trast to volume average), is shown to be more generally applicable to
calculate the effective parameters along both 0X and 0M directions.
Along the 0X and 0M directions, we find that it is convenient to
introduce κ
eff
=(c
eff
11
+c
eff
12
)/2 and µ
eff
=(c
eff
11
−c
eff
12
)/2 as the effective
elastic bulk modulus and shear modulus that correspond with
the monopolar and quadrupolar resonances, respectively. Whereas
ρ
eff
= −F
eff
x
/ω
2
u
eff
x
a
2
, where both the effective force F
eff
x
on the
unit cell and its effective displacement u
eff
x
may be obtained from
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: T
eff
xx
=c
eff
11
S
eff
xx
+c
eff
12
S
eff
yy
,
T
eff
yy
= c
eff
12
S
eff
xx
+c
eff
11
S
eff
yy
, and T
eff
xy
= 2c
eff
44
S
eff
xy
, where both the effective
stresses and the effective strains are evaluated on the unit cell
boundary. Details can be found in the Supplementary Information.
In the lower negative band, the µ
eff
evaluated from the relevant
eigenstates is negative and diverges (in the negative direction) at
the 0 point owing to the quadrupolar resonance, whereas κ
eff
is
622 NATURE MATERIALS  VOL 10  AUGUST 2011  www.nature.com/naturematerials
© 2011 Macmillan Publishers Limited. All rights reserved
Page 3
NATURE MATERIALS DOI: 10.1038/NMAT3043
ARTICLES
positive and finite. In the higher negative band, κ
eff
is negative
and diverges (in the negative direction) at the 0 point owing to
the monopolar resonance, whereas µ
eff
is positive and finite. The
compressional wave and shear wave velocities along the 0X (0M)
direction can be obtained by
√
(κ
eff
+µ
eff
)/ρ
eff
(
√
(κ
eff
+c
eff
44
)/ρ
eff
)
and
√
c
eff
44
/ρ
eff
(
√
µ
eff
/ρ
eff
), respectively. From these formulas, it
can be clearly seen that in the higher frequency negative dispersion
band, the fact that κ
eff
0, ρ
eff
< 0, µ
eff
> 0 and c
eff
44
> 0 implies
only the longitudinal wave can propagate along the 0X and 0M
directions. For the lower frequency negative dispersion band we
have µ
eff
0, ρ
eff
< 0 and other parameters positive, therefore
only the longitudinal (transverse) wave can propagate along the 0X
(0M) 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, c
eff
44
for the lower negative band cannot
be obtained from the effective medium theory. Thus the condition
c
eff
44
> 0 is inferred by consistency with the band structures.
Discussion of novel characteristics
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
perform transmission calculations by using COMSOL Multiphysics
to illustrate the transmission properties of the hybrid bands. A
schematic for the numerical setup in COMSOL is shown in Fig. 4a.
For a slab of metamaterial with seven layers along the 0M direction,
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.
The transmissions for the longitudinal or transverse waves (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 0M 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
the obtained transmissions along the 0X direction under transverse
and longitudinal inputs, respectively. Large transmissions are seen
only for longitudinal input excitations, for both negative dispersion
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
(normalization issues) do not affect the analysis here.
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
degenerate modes at the 0 point (with displacements marked as red
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 0X direction and longitudinal in
the 0M direction, whereas the other mode (blue arrows) is just the
opposite. When the steel rods are inserted at the positions of the
dashed boxes, the symmetry of the rubber rod is broken and the two
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 metamaterials
30–32
. During the effective medium
calculation of the lower negative band, we have actually observed
stresses T
eff
xy
6=T
eff
yx
, which implies ‘local rotations’ beyond the linear
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 T
eff
xx
and T
eff
yy
, as can be seen from the divergence of
κ
eff
and µ
eff
in Fig. 2b and c, respectively. Nevertheless, the ‘local
rotation’ effect may have some interesting implications in other
types of metamaterials
30–32
.
The ‘fluidlike’ and ‘superanisotropic’ hybrid elastic solids
shown in this report represent two types of new solids absent in
nature. They significantly extend our ability to control elastic waves
in solids. Potential applications of these hybrid elastic solids include
wave polarizers, wave imaging and confinement
33
, controlling
elastic and seismic waves
34,35
, transformation acoustics
36
, and so
on. As a result of their doublenegative nature, negative refraction
and superlensing for longitudinal or transverse components are also
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
References
1. Pendry, J. B., Holden, A. J., Robbins, D. J. & Stewart, W. J. Magnetism from
conductors and enhanced nonlinear phenomena. IEEE Trans. Microwave
Theory Tech. 47, 2075–2084 (1999).
2. Shelby, R. A., Smith, D. R. & Schultz, S. Experimental verification of a negative
index of refraction. Science 292, 77–79 (2001).
3. Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and negative
refractive index. Science 305, 788–792 (2004).
4. Fang, N., Lee, H., Sun, C. & Zhang, X. Subdiffractionlimited optical imaging
with a silver superlens. Science 308, 534–537 (2005).
5. Soukoulis, C. M., Linden, S. & Wegener, M. Negative refractive index at optical
wavelengths. Science 315, 47–49 (2007).
6. Lezec, H. J., Dionne, J. A. & Atwater, H. A. Negative refraction at visible
frequencies. Science 316, 430–432 (2007).
7. Valentine, J. et al. Threedimensional optical metamaterial with a negative
refractive index. Nature 455, 376–379 (2008).
8. Yao, J. et al. Optical negative refraction in bulk metamaterials of nanowires.
Science 321, 930 (2008).
9. Veselago, V. G. The electrodynamics of substances with simultaneously
negative values of ε and µ. Sov. Phys. Usp. 10, 509–514 (1968).
10. Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85,
3966–3969 (2000).
11. Leonhardt, U. Optical conformal mapping. Science 312, 1777–1780 (2006).
12. Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields.
Science 312, 1780–1782 (2006).
13. Pendry, J. B. & Ramakrishna, S. A. Nearfield lenses in two dimensions.
J. Phys. Condens. Matter 14, 8463–8479 (2002).
14. Yang, T., Chen, H. Y., Luo, X. D. & Ma, H. R. Superscatterer: Enhancement of
scattering with complementary media. Opt. Express 16, 18545–18550 (2008).
15. Lai, Y., Chen, H. Y., Zhang, Z. Q. & Chan, C. T. Complementary media
invisibility cloak that cloaks objects at a distance outside the cloaking shell.
Phys. Rev. Lett. 102, 093901 (2009).
16. Lai, Y. et al. Illusion optics: The optical transformation of an object into
another object. Phys. Rev. Lett. 102, 253902 (2009).
17. Liu, Z. et al. Locally resonant sonic materials. Science 289, 1734–1736 (2000).
18. Liu, Z., Chan, C. T. & Sheng, P. Analytic model of phononic crystals with local
resonances. Phys. Rev. B 71, 014103 (2005).
19. Yang, Z., Mei, J., Yang, M., Chan, N. H. & Sheng, P. Membranetype acoustic
metamaterial with negative dynamic mass. Phys. Rev. Lett. 101, 204301 (2008).
20. Fang, N. et al. Ultrasonic metamaterials with negative modulus. Nature Mater.
5, 452–456 (2006).
NATURE MATERIALS  VOL 10  AUGUST 2011  www.nature.com/naturematerials 623
© 2011 Macmillan Publishers Limited. All rights reserved
Page 4
ARTICLES
NATURE MATERIALS DOI: 10.1038/NMAT3043
21. Ding, Y. Q., Liu, Z. Y., Qiu, C. Y. & Shi, J. Metamaterial with simultaneously
negative bulk modulus and mass density. Phys. Rev. Lett. 99, 093904 (2007).
22. Li, J. & Chan, C. T. Doublenegative acoustic metamaterial. Phys. Rev. E 70,
055602 (2004).
23. Lee, S. H., Park, C. M., Seo, Y. M., Wang, Z. G. & Kim, C. K. Composite
acoustic medium with simultaneously negative density and modulus.
Phys. Rev. Lett. 104, 054301 (2010).
24. Li, J., Fok, L., Yin, X. B., Bartal, G. & Zhang, X. Experimental demonstration of
an acoustic magnifying hyperlens. Nature Mater. 8, 931–934 (2009).
25. Zhang, S., Yin, L. L. & Fang, N. Focusing ultrasound with an acoustic
metamaterial network. Phys. Rev. Lett. 102, 194301 (2009).
26. Royer, D. & Dieulesaint, E. Elastic Waves in Solids (Springer, 1999).
27. Tamura, S & Wolfe, J. P. Coupledmode stop bands of acoustic phonons in
semiconductor superlattices. Phys. Rev. B 35, 2528–2531 (1987).
28. Zhou, X. M. & Hu, G. K. Analytic model of elastic metamaterials with local
resonances. Phys. Rev. B 79, 195109 (2009).
29. Wu, Y., Lai, Y. & Zhang, Z. Q. Effective medium theory for elastic metamaterials
in two dimensions. Phys. Rev. B 76, 205313 (2007).
30. Milton, G. W. New metamaterials with macroscopic behaviour outside that of
continuum elastodynamics. New J. Phys. 9, 359 (2007).
31. Milton, G. W. & Willis, J. R. On modifications of Newton’s second law and
linear continuum elastodynamics. Proc. R. Soc. A 463, 855–880 (2007).
32. Willis, J. R. The nonlocal influence of density variations in a composite.
Int. J. Solids Struct. 210, 805–817 (1985).
33. Guenneau, S., Movchan, A., Petursson, G. & Ramakrishna, S. A. Acoustic
metamaterials for sound focusing and confinement. New J. Phys. 9, 399 (2007).
34. Brun, M., Guenneau, S. & Movchan, A. B. Achieving control of inplane elastic
waves. Appl. Phys. Lett. 94, 061903 (2009).
35. Farhat, M., Guenneau, S. & Enoch, S. Ultrabroadband elastic cloaking in thin
plates. Phys. Rev. Lett. 103, 024301 (2009).
36. Chen, H. Y. & Chan, C. T. Acoustic cloaking and transformation acoustics.
J. Phys. D 43, 113001 (2010).
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.
624 NATURE MATERIALS  VOL 10  AUGUST 2011  www.nature.com/naturematerials
© 2011 Macmillan Publishers Limited. All rights reserved
Page 5
Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

 "The study of the elastodynamics of heterogeneous materials has led to the discovery of a special class of a composite [1]. They exhibit remarkable acoustic properties ranging from near zero trans missibility [2], enhanced absorption [3], negative dynamic mass density [4]/ bulk modulus [5], negative refractive index [6], super anisotropy, zero rigidity [7] etc. These exotic phenomena have numerous potential applications such as low frequency noise attenuation [3], isolation of civil structures from seismic waves [8], superlenses with a resolution beyond the Rayleigh limit [6,9] , waveguides that can be used to channel acoustic waves, etc. "
Article: Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum
[Show abstract] [Hide abstract] ABSTRACT: This contribution presents a novel homogenization technique for modeling heterogeneous materials with microinertia effects such as locally resonant acoustic metamaterials. Linear elastodynamics is used to model the micro and macro scale problems and an extended first order Computational Homogenization framework is used to establish the coupling. Craig Bampton Mode Synthesis is then applied to solve and eliminate the microscale problem, resulting in a compact closed form description of the microdynamics that accurately captures the Local Resonance phenomena. The resulting equations represent an enriched continuum in which additional kinematic degrees of freedom emerge to account for Local Resonance effects which would otherwise be absent in a classical continuum. Such an approach retains the accuracy and robustness offered by a standard Computational Homogenization implementation, whereby the problem and the computational time are reduced to the online solution of one scale only. 
 "Steel and foam are selected to act as " masses " and " springs " , respectively. The material parameters are density q ¼ 7784 kg/m 3 , Young's modulus E ¼ 207 GPa, and Poisson's ratio 0.3 for steel [25]; and density q ¼ 115 kg/m 3 , Young's modulus E ¼ 8 MPa, and Poisson's ratio 0.33 for foam [12]. A fixed cylindrical steel shell connects the foam externally, which acts as the " ground " and increases the stiffness of the whole system. "
Article: Elastic Metamaterials With LowFrequency Passbands Based on Lattice System With OnSite Potential
[Show abstract] [Hide abstract] ABSTRACT: An elastic metamaterial with a lowfrequency passband is proposed by imitating a lattice system with linear onsite potential. It is shown that waves can only propagate in the tunable passband. Then, two kinds of elastic metamaterials with double passbands are designed. Great wave attenuation performance can be obtained at frequencies between the two passbands for locally resonant type metamaterials, and at both low and high frequencies for the diatomic type metamaterials. Finally, the strategy to design twodimensional (2D) metamaterials is demonstrated. The present method can be used to design new types of smallsize waveguides, filters, and other devices for elastic waves. 
 "Much effort has been devoted in recent years to study of the propagation of acoustic/elastic waves in the periodic composites called PCs12345. PCs are artificial media consisting of periodic inclusions in a matrix background with various topologies678910. They can exhibit various special physical properties such as phononic band gaps in which waves are prevented from propagating. "
[Show abstract] [Hide abstract] ABSTRACT: Band structures are investigated in twodimensional phononic crystals (PC) composed of a periodic Sshaped slot in an air matrix with a square lattice. Dispersion relations, pressure fields and transmission spectra are calculated using the finite element method and Bloch theorem. Numerical results show that the proposed PC can yield complete and large band gaps at lower frequency ranges compared with that of the Jerusalem slots in Li et al. (Phys B 456:261–266, 2015) under the same parameter setting of the lattice and outline of the inclusions. The transmission spectrum is verified to be reasonably consistent with the band gaps along the \(\Gamma X\) direction. By analysing the pressure fields of several modes, the resonance modes of cavities within the Sshaped slot structure are found to result in the lowfrequency band gaps. The effects of the geometrical parameters on the upper and lower edges of the first and second complete band gap are further studied. Numerical results show that the bandwidth of the first and second band gaps can be modulated over an extremely large frequency range by the geometrical parameters. The properties of the proposed PC have potential for implementation in structures and devices of noise and vibration control, such as noise filters and waveguides.