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Andrew Norris Rutgers University
Metamaterials in Acoustics and Vibration:
From Theory to Practice
Rayleigh Lecture
ASME NCAD IMECE 2011
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concepts
geometry = material properties via homogenization or transformation or both Applications: narrow band : phononic crystals, negative index materials broadband: acoustic cloaking
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
overview
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introduction
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Word first used in 1999 by Rodger Walser of the University of Texas, Austin. Metamaterials are:
negative index of refraction microwave cloak
Macroscopic composites having a manmade, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation.
wiki/metamaterial
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normal behavior
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Pendry, Schurig and Smith Science 2006
A cloak makes waves travel around an object
normal behavior
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Metamaterial Electromagnetic Cloak at Microwave Frequencies Schurig, Mock, Justice, Cummer, Pendry, Starr, Smith, Science, 2006
simulation perfect cloak imperfect cloak experiment copper cylinder cloaked cylinder
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why now?
combination of: • cross-disciplinarity • advanced fabrication techniques • computational availability • thinking “outside the box”
Why should engineers be interested in metamaterials?
potential for radically new devices/technologies - vibration insulation - ultrasonic imaging beyond the diffraction limit - control of SAWs: filtering, guiding, … - stealth for underwater structures
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
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phononic crystals
engineering the band gap
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phononic crystals periodic mechanical systems
phononic band gaps provide tools for controlling waves:
filtering, steering, trapping, focusing beyond the diffraction limit, etc.
2-D
periodic in two directions
3-D
periodic in three directions
1-D
periodic in one direction
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acoustics
pressure relation
momentum balance
assume constant
wave speed
water 1500 m/s
air 330 m/s
density
bulk modulus
ratio of pressure to particle
velocity in a plane wave acoustic impedance
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scattering of sound
planewave
)(~ txkiep
k / c 2
k
scattering
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periodic system
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
for most , sound propagates through crystal without scattering (scattering cancels coherently)
...but for some (~ 2a), no sound propagates: a phononic band gap
a
planewave
)(~ txkiep
k / c 2
k
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Introduction: metamaterials
Phononic crystals – engineering the band gap
waves in periodic systems
examples of phononic crystals
Acoustic cloaking – engineering the impossible
transformation acoustics
1D, 2D, cylindrical
inertial materials
Pentamode materials
cloaking elastic waves
Metatheory for metamaterials?
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waves in periodic systems G. Floquet (1883)
Bloch-Floquet theorem applies:
)(),x( xetk
txki
Uu
periodic envelope plane wave
k is constant, i.e. no scattering of Bloch wave
given by the finite unit cell,
so the frequencies are discrete n(k) kU
F. Bloch
F. Bloch (1928)
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band gap
waves in periodic systems: 1D
r1 r2 r1 r2 r1 r2 r1 r2 r1 r2 r1 r2
r(x) = r(x+a) a
k is periodic:
k + 2π/a equivalent to k
k
0 π/a –π/a
irreducible Brillouin zone
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Any 1D periodic system has a gap
r1
r(x) = r(x+a) a
k
0 π/a –π/a
first, treat as
artificially periodic
bands are “folded”
by 2π/a equivalence
xa
xa
eex
aix
ai
sin,cos
,
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r(x) = r(x+a) a
r1
0 π/a
sin
ax
cos
ax
x = 0
still treat it as “artificially” periodic
forward, backward propagating waves
or standing waves
Any 1D periodic system has a gap
k
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band gap
Any 1d periodic system has a gap
0 π/a
add a small “real”
density periodicity
r2 = r1 + Dr
sin
ax
cos
ax
r(x) = r(x+a) a
r1 r2 r1 r2 r1 r2 r1 r2 r1 r2 r1 r2
x = 0
splitting of degeneracy: wave mainly in slower zone (r2)
has lower frequency
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Rayleigh predicted band gaps in 1887
Rayleigh 1870
1885
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Bloch waves – 1D periodic system
time harmonic
r(x) = r(x+a) a
r1 r2 r1 r2 r1 r2 r1 r2 r1 r2 r1 r2
x = 0
propagator
two phase medium
Bloch condition
dispersion relation
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Introduction: metamaterials
Phononic crystals – engineering the band gap
waves in periodic systems
examples of phononic crystals
Acoustic cloaking – engineering the impossible
transformation acoustics
1D, 2D, cylindrical
inertial materials
Pentamode materials
cloaking elastic waves
Metatheory for metamaterials?
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phononic crystals
example applications
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filtering and controlling waves
perfect mirror (in band gap)
- frequency filter
wave guide
- beam splitting, multiplexing
Khelif et al (APL 2004)
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phononic crystals for surface acoustic waves
Mohammadi et al. (APL 2008)
cubic arrays give larger gaps than triangular
but are not as isotropic
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partial band gap for waves in some directions only
Complete band gap - all directions
stronger scattering leads to greater band gaps
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resonant oscillator metamaterials
H. H. Huang, C. T. Sun, New J. Phys. 11 2009
mass-in mass system
equivalent system
Negative effective mass for
Compare equilibrium eqns:
Control bandgap frequencies
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low frequency band gap: heavy & soft Lead coated spheres + layer of silicone rubber in cubic array
Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng (Science 2000)
sub-wavelength band gap λ >> a
very low frequency band gap and negative elastic constant due to dipole resonance
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negative effective modulus
N. Fang et al. (2006 Nature Mater. 5 452-6)
unit cell = Helmholtz resonator
effective modulus:
V = cavity vol, L’ = neck length, S= neck X-section
negative effective modulus gives negative group velocity leading to group delay
periodic array
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Sánchez-Dehesa, Torrent, L-W Cai (NJP 2009)
• uses multiple scattering theory (MST)
2D sonic crystals with tailored properties
lensing effect plane wave scattering from finite array of regularly spaced identical cylinders equivalent material with radially varying sound speed
gradient index lens
plane wave focusing
expt MST
Martin et al. (APL 2010)
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mechanism of sonic crystal focusing
dense solid cylinders in lighter fluid leads to effective medium with
at the same time the effective impedance is not much different
“normal” lens effect
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phononic crystals can lead to negative refraction
potential for resolution beyond the diffraction limit
“perfect lens”
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ultrasonic negative index lens at 0.55 MHz
negative index for sound in water: fluid matrix NIM
Sukhovich, Jing, Page (PRB 2008)
phase matching to negative group velocity
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Morvan et al. APL 2010
solid matrix negative index materials
Metal matrix
Epoxy matrix Croenne et al. PRB 2011
Tuning properties to efficiently couple with sound in water remains a challenge
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phononic crystals in engineering applications
acoustic insulation e.g. sound barriers using inexpensive materials low frequency vibration isolation, using internal resonators waveguides, SAW filters great potential, possible to fabricate, but radio-frequency devices still limited by energy loss negative index materials, lens work well in theory could provide super focusing, e.g. biomedical imaging - not yet practical: impedance mismatch, material limits
(Lin et al. JAP ’09)
(Croenne et al. ’11)
(Sanchez-Dehesa et al. ’10)
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minimalist sculpture by E. Sempere (1923-1985) in a Madrid park, was demonstrated to be a phononic crystal in 1995
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
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cloaking engineering the impossible
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Electromagnetic equations are invariant under transformation of coordinates - old idea - singular transformations yield cloaking - 2D cloaking demonstrated at microwave frequency Schurig et al ('06)
Acoustic -acoustic cloaking 2D and 3D models using singular transformation Chen & Chan ('07), Cummer et al ('07, '08) -these models are restricted to anisotropic inertia -general theory allows pentamode material (Norris ('08, '09) practical demonstration of: broadband ultrasonic cloak (Zhang et al., PRL 2010) acoustic carpet cloak (Popa et al., PRL 2011)
background
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broadband cloaking demonstration S. Zhang, C. Xia and N. Fang (PRL 2011)
2D transmission line approach uses sub-wavelength acoustic Helmholtz resonator
60 kHz
no cloak
with cloak
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Acoustic carpet cloak in air
Popa, Zigoneanu, Cummer (PRL June 2011)
Idea of “carpet cloak”: cover the body so it looks like there is nothing there
theory/simulation experiment at sonic frequency
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
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cloaking transformation acoustics
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b
same impedance/ no reflection same travel time
1D acoustic cloak mirage/illusion
a
“cloaked” region metafluid
idea of transformation: geometrical mapping defines the material properties
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same impedance/ no reflection same travel times
transformation material properties
“cloaked” region
b 1
b 2
a 1
a 2
total mass is conserved
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scattering of waves
= reflection in all directions
much harder !
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2D acoustic mirage
a b
impedance travel time Snell Idea works at only one angle of incidence ???
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finite transformation (deformation)
Change of coordinates/Laplacian in new variables:
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transformation of the acoustic wave equation
equation transforms if the density in the deformed (current) description is anisotropic
= idea behind transformation acoustics
identical to if
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
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anisotropic inertia ?
momentum balance with anisotropic inertia tensor
standard acoustic wave equation
standard pressure constitutive relation
modified acoustic wave equation e.g. layered fluid exhibits anisotropic inertia
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2D mirage using anisotropic inertia
a b
works for all angles of incidence
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Acoustic carpet cloak in air
Popa, Zigoneanu, Cummer (PRL June 2011)
How?
Liang & Li (APL 2011)
layered scaffold of heavy plates -has effect of slowing & curving the wave so that it appears to reflect from the flat surface - curved 2D mirage device - carpet cloak in water using curved steel plates
physical region is mapped into larger virtual one
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Layman et al. (APL 2011 doi: 10.1063/1.3652914)
using transformation acoustics to design wave control devices
example: cylindrical-to-plane wave lens
virtual physical
range of materials used cylindrical source radiates as plane waves
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Metamaterial Electromagnetic Cloak at Microwave Frequencies Schurig, et al., Science, 2006
perfect cloak imperfect cloak copper cylinder cloaked cylinder
extremely anisotropic
anisotropy is a big part of cloaking
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aside: strange effects in cloaking - rays in a cloak
the wavefront bifurcates the line through the singular point becomes a split ray the “uncaustic” ray bundle density
rays = deformed straight lines under transformation F
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Momentum Balance
Torrent and Sanchez-Dehesa (2008)
cylindrical anisotropic inertia
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homogenized cylinders with anisotropic inertia
effective properties
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application: acoustic hyperlens
Li et al. (Nature Materials 2009) doi: 10.1038/nmat2561
images sub-wavelength sources
sources
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cloaking : metafluid + hole
r = b R= b
virtual / original physical / transformed
physical virtual
transformation
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radially symmetric cloak
blow up R=0 into ball of radius
The perfect cloak has inner boundary equal to the image of a point In a near cloak it is a small hole imperfect or “near cloak” perfect (almost)
example transformation:
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three-fluid inertial cloak (Norris and Nagy, JASA 2010)
effective compressibility:
ODE for R(r) explicit transformation r → R(r) Bottom line: can achieve exact transformation using only 3 distinct fluids mixed in different proportions
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example: rigid scatterer with 3-fluid near-cloak
cylinder sphere
● can reduce scattering cross-section to less than 1%
● requires one very heavy fluid, one very light
● total mass of device is LARGE
● BUT – the types of fluid required are not available e.g. huge density + huge compressibility
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near cloaks made of inertial metafluids have very large mass
total mass of 3D near-cloak
The perfect cloak has infinite mass – regardless of the transformation (2D or 3D, radially symmetric or not)
???
perfect cloak:
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How to make a cloak? Acoustic ``Invisibility" Cloaks Possible, Study Says National Geographic 1/08 How to Make a Submarine Disappear Science/AAAS 1/2008 Acoustic metamaterials: Silence all around Nature/China 11/07 How to Build an Acoustic Invisibility Cloak New Scientist 8/07 etc.
All inertial cloaks - infinitely massive
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Path to acoustic cloaking?
Transformation -> radially anisotropic density -> cloak -> zero scattering
Making a fluid with strong density anisotropy is difficult/impossible. Other issues: total mass required is very large
Why density? Why not bulk modulus?
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
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cloaking pentamode materials
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water as an elastic “solid”
Kelvin (1856)
Acoustics Five of the eigen-stiffnesses are zero. Water is a pentamodal elastic material (Milton & Cherkaev 1994) five (penta) easy modes
elastic equation of motion
general pentamode form of stiffness:
S is divergence free
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pentamode material = transformed acoustic medium
constitutive relation momentum balance
rewrite in 'acoustic' form:
Use the PM compatibility condition
wave equation for the pseudo-pressure
(was )
pseudo-pressure p(x,t)
Can transform back to acoustic eq in X for arbitrary S as long as it satisfies
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why can we have different types of transformed materials?
started with transformed coordinates interpreted as eq. of acoustic fluid with anisotropic inertia but we did not change the meaning of the pressure (or particle displacement) relax this and redefine a “transformed” displacement
U = A u
then A is related to appearance of the matrix S in the pentamode material
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mechanical behavior of pentamode materials a single type of stress (and strain) - generalize hydrostatic stress and volumetric strain of an acoustic fluid
Static equilibrium of a block ... under gravity
g
“microstructure”
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1) most general type of acoustic metafluid is PM with anisotropic inertia 2) can get isotropic density by choosing S to make it so
The metafluid is any material of the form *
- J (Jacobian) and V (metric) are defined by the (arbitrary) transformation - S is any definite symmetric tensor satisfying anisotropic inertia is the special case S = I S introduces additional degrees of freedom, non-uniqueness (unlike EM)
summary of pentamode material transformation
*it can be even more general: S does not have to be symmetric - leading to nonsymmetric stress
(Norris ’08, ’09)
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- looks a lot like the inertial anisotropy transformation: - gives the same wave steering effects
- but the mechanics is completely different
pentamode material transformation
radially symmetric transformation
r R
physical virtual
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freedom to choose S means the cloak material has isotropic inertia e.g. for any radially symmetric transformation Important PM property - like 1D (used later): The total mass of the cloak is conserved under the transformation
a non-radially symmetric cloak with isotropic density (& finite mass of course)
transformed PM has isotropic inertia
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simulation: acoustic scattering from a steel sphere
no cloak layered pentamode cloak
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials - design cloaking elastic waves Metatheory for metamaterials?
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design of pentamode materials
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possible designs of acoustic cloaking devices
force/truss structure for 2D cloak Idea: unit cell supports only stress S
1) anisotropic inertia via fine layering
● highly constrained ● large mass issues ● excessive material properties
2) Pentamode Material ● isotropic inertia ● solid based ● microstructure ● use ideas from composite materials
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acoustic fluid
Pentamode Material (possible) microstructures
Fluid only supports stress -p I
PM supports only stress S Idea: create unit cell that supports S (elements indicate forces)
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isotropy
anisotropy
isotropic and anisotropic networks
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a generic material?
some desired/necessary properties: • transparent to underwater sound in its “base state”
• density of water in its base state (conservation of mass property)
• effective bulk modulus of water in base state
• “deformable” to achieve anisotropy
• maximize potential cloaked space
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isotropic anisotropic
pentamode material (PM) microstructure possibilities
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cellular foam structures
effective elastic moduli
extensional and bending compliances (inverse of stiffness)
(Kim & Hassani 2003)
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thin members have small bending stiffness, leading to (approximately)
Cellular structure has PM form with unique stress
acoustic fluid:
bottom line: cellular foam-like structure with thin members can provide the PM stiffness with relatively small shear rigidity
thin members = more potential cloaking space ……….. stiff, dense
cellular foam as a Pentamode Material
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Metal Water
Idea: make water from metal foam e.g. start with block of Al, remove solid metal to get: a) effective density of water b) effective isotropic elasticity with bulk modulus of water, small shear modulus
Vacuum (air)
metal (e.g. aluminium)
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Have manufactured prototypes from Aluminum plate stock using a water-jet process.
In-water transmission testing of the full part completed.
MW prototypes
Bulk modulus = 2.25 Gpa Density = 1000 kg/m^3
Shear modulus = 0.065 GPa
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FEM
islands : inertial role only - denser is better isthmuses : elastic role only - stiffer is better
metal
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Eigenanalysis of the Metal Water design
indicates shear-related modes.
As expected, since the design cannot be and is
no perfectly Pentamode.
Shear modes are high-k, low-frequency, easily
damped, not expected to affect test results.
Transient explicit analysis of the Metal Water
design indicates that the shear-related
deformation modes do not have a strong effect
on wave propagation, when a sample is
ensonified using a plane wave.
FLEX simulations (Weidlinger Associates)
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Metal Water cloak
New metafluid = radially compressed version of metallic water same total mass as uncompressed region Conservation of cloaked space
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conservation of mass = conservation of cloaked space
Heavy metal preferred
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Metal Water offers the potential of solid structures that • mimic the acoustic properties of water (isotropic) • enable transformation acoustics (anisotropic)
using the long wavelength homogenization properties of structured metallic foams as Pentamode materials Other uses?
PM – Summary
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Dispersion curve in the first Brillouin zone
G X
FR
EQ
UE
NC
Y (
MH
z)
0.00
0.01
0.02
0.03
0.04
X J
dentelle de Norris - 30°
0,00E+00
5,00E+03
1,00E+04
1,50E+04
2,00E+04
2,50E+04
3,00E+04
3,50E+04
4,00E+04
0 40 80 120 160 200
Nombre d'onde (1/m)
Fréq
uenc
e (H
z)
Sound in water
Metal Water: application to negative index materials
material properties appear to match with sound in water Goal: NIM lens with MW using negative group veocity branches
negative group velocity branch
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
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cloaking of elastic waves
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cloaking in elasticity ?
miraging and cloaking, work in principle - but require materials that are not “elastic”
- Density can always be made isotropic - Stress is usually not symmetric, Cosserat materials are necessary - normal elastic materials can provide approximate cloaking
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elastic transformation theory
(ANN, ALS 2011)
• transformation
• as in acoustics, the materials are not unique. They can be characterized by how the displacement transforms
• 2 parameters:
• material generally of Willis form, with properties including matrix density that are functions of frequency
• BUT constant isotropic density if then moduli of Cosserat form
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elastic transformation unique nonlinear materials
transformed elastic moduli are tangent moduli for a hyperelastic material with strain energy under a state of (pre)stress
(ANN, ALS 2011)
• equilibrium of the pre-stress constrains the transformation F to satisfy
• for acoustics, this includes all radially symmetric (d=2,3) transformations
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this means elastic waves can be cloaked using a material with a unique finite strain energy function expand small holes to finite size: the deformed solid has small-on-large moduli exactly those required from the transformation
ex. SH waves
bulk material initial hole expanded
line source - rigid cylinder - no cloak pressurize small hole to size of cylinder, and insert rigid body scattering from original hole
neo-Hookean strain energy function = original shear modulus
cloaking in elasticity
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Introduction: metamaterials Phononic crystals – engineering the band gap waves in periodic systems examples of phononic crystals Acoustic cloaking – engineering the impossible transformation acoustics 1D, 2D, cylindrical inertial materials Pentamode materials cloaking elastic waves Metatheory for metamaterials?
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A metatheory for metamaterials?
Phononic crystals:
homogenization of periodic systems at finite frequency & wavelength leads to effective material properties consistent with the Willis constitutive model: (Willis 2010, Milton and Willis 2006, Shuvalov et al. 2010, Srivistava & Nemat-Nasser 2011)
Acoustic and elastic cloaking:
The most general constitutive model that is invariant under “transformation” is the Willis constitutive model. (Milton, Briane, Willis, 2007, ANN 2011)
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metamaterials, especially cloaking devices, combine a rich mixture of topics • wave mechanics/physics • continuum mechanics • differential geometry • anisotropic elasticity • finite elasticity • materials science • fabrication issues • computational methods • etc.
in closing
![Page 106: Cloaking of acoustic and elastic wavesfiles.asme.org/Divisions/NCAD/30914.pdfcopper cylinder cloaked cylinder extremely anisotropic anisotropy is a big part of cloaking aside: strange](https://reader033.vdocuments.us/reader033/viewer/2022060916/60a9b48eb634400eb37e1078/html5/thumbnails/106.jpg)
Thanks to:
A. Nagy Rutgers J. Cippola Weidlinger N. Gokhale Weidlinger G. Orris NRL A-C Hladky-Hennion IEMN/Lille W. Parnell U. Manchester A. Shuvalov U. Bordeaux 1
ONR, NSF