NATURE MATERIALS DOI: 10.1038/NMAT3043
positive and finite. In the higher negative band, κ
and diverges (in the negative direction) at the 0 point owing to
the monopolar resonance, whereas µ
is positive and finite. The
compressional wave and shear wave velocities along the 0X (0M)
direction can be obtained by
), respectively. From these formulas, it
can be clearly seen that in the higher frequency negative dispersion
band, the fact that κ
< 0, µ
> 0 and c
> 0 implies
only the longitudinal wave can propagate along the 0X and 0M
directions. For the lower frequency negative dispersion band we
< 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
for the lower negative band cannot
be obtained from the effective medium theory. Thus the condition
> 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 ‘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
perform transmission calculations by using COMSOL Multiphysics
to illustrate the transmission properties of the hybrid bands. A
schematic for the numerical set-up 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 ‘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
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 ‘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 metamaterials
. During the effective medium
calculation of the lower negative band, we have actually observed
, 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
, as can be seen from the divergence of
in Fig. 2b and c, respectively. Nevertheless, the ‘local
rotation’ effect may have some interesting implications in other
types of metamaterials
The ‘fluid-like’ and ‘super-anisotropic’ 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
elastic and seismic waves
, transformation acoustics
, and so
on. As a result of their double-negative 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
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