Andrew Norris
Rutgers UniversityNJ, USA
Transformation Acoustics, Pentamode Lenses
and Spece-Time Modulated Phononic Crystals
1
Novel Optical Materials
14 March 2017
2
• Transformation Acoustics
Nonuniqueness
Pentamode materials
Examples of Pentamode lenses with different Mos
Conformal Transformation Acoustics
Highly directional TA lenses
• Space-time modulated (activated) phononic crystal
Exact result for wave-like modulation
Low frequency limit: Willis equations
bit.do/acousticmetamaterials
Alexey Titovich Rutgers
Adam Nagy ..
Xiaoshi Su ..
Jeffrey Cipolla Weidlinger
A.C. Hladky-Hennion Lille/IEMN
M.R. Haberman ARL/UT
C. Cushing ..
Hussein Nassar U. Missouri
Guoliang Huang ..
1. Slower
2. Matched impedance
Same total mass
& overall compressibility
Transformation acoustics
How to make an illusion?
3
Works for all incidence directions
speed depends on direction
a ousti a isotropy
The transformation material properties
4
unchanged speed
assume stiffness & density are anisotropic
1D results
Horizontal
speed
Transformation acoustics nonuniqueness
One solution, keep K’ isotropic :
4 parameters, 3 equations : 1 degree of freedom
another solution - isotropic density : 5
water
PM
anisotropic inertia or anisotropic stiffness
Pentamode materials
cloaked
regioncloaked
region
Norris Proc. R. Soc. Lond. A 2008
acoustically transformed materials are not unique
The same transformation can be achieved with different metamaterials
- huge difference from electromagnetics
6
pentamode material is the limiting case of anisotropic elastic solids with zero shear rigidity
Milton & Cherkaev 1995
Inertial vs. Pentamode cloaking
Inertial :
layers of fluidscloak
Pentamodecloak has the
Same mass as the region
it cloaks
Pentamode:
lattice structure
Inertial cloak has very large
mass - infinite if perfect
Norris Proc. R. Soc. Lond. A 2008, JASA 2009
7
Two examples of Pentamode focusing devices
60 mm
265 mm
Negative index
phononic crystal
TA/Gradient index lens
bulk modulus = 2.25 GPa
density = 1000 kg/m3
shear modulus = 0.065 GPa(i.e. small)
homogenize
Metal Water pentamode structure
9
sonic
lines
71.2 kHz
71.26kHz
70.8 kHz
negative
refraction
(A-C Hladky-Hennion)
One wave
region
negative
Refraction?
ImageSource
F = 79.5 kHz
F = 80 kHz
F = 81kHz
sourceimage
on a line // to the slab
Hladky-Hennion et al. 2013, 2014
simulation
MW : SOLID NEGATIVE INDEX LENS
10Measurements at IEMN, Lille
11
Metal Water structure
30 kHz Source
60 mm
265 mm
80 mm
A-C Hladky-Hennion et al.
sonic
lines
12
Pentamode architecture
• Expands available index + impedance range
• Gradient properties
• Frequency range depends on unit cell size
Homogenized properties.
Relative to water
Pentamode metal structures with low shear can achieve significant
effective anisotropy but also a wide range of isotropy
Pentamode acoustic elements
Feb. 23 - 24, 2016 13
• Aluminum pentamode structure provides high effective speeds
• Impedance matched
• Broadband
• Low a erratio Tra sfor atio A ousti s odified hyper oli se a t profile
Hyperbolic secant profile after coordinate stretch
20 kHz 30 kHz
10 cm
Pentamode underwater GRIN lens
stretch
40 cm
13.7 cm
Gradient index lens with Metal Water
Measurements at Applied Research Laboratories, UT Austin
by Michael Haberman and Colby Cushing
Gradient index lens with Metal Water
Plane wave focusing
35 kHz
Sonic line
one wave region
16
3D Pentamode in water
Bounding elastic plates connects exterior acoustic fluid to the pentamode material
elastic plates
Far-Field
Directivity
Near-Field
Pressure
27.5 kHz
-10
-5
0
5
10
30
210
60
240
90
270
120
300
150
330
180 0
27.5 kHz
Example of expected response
0
0.5
1.0
1.5
2.0
2.5
Monopole to directional source lenses
17
GOAL: Convert monopole source to collimated beams
Conformal Transformation Acoustics
Conformal mappings preserves isotropy
Norris APL 2012
18
Density is CONSTANT
Provides exact basis for gradient index lens design
E.G. circle to rectangle mapping
Energy flux density
Rays
19
20.5 kHz
21.4 kHz
2 cm
Hexagonal pentamode unit cells
32.5 cm
thinnest beam: 0.29 mm
Pentamode cylindrical-to-plane wave lens
Monopole source
Circle to Square lens: Array of Tubes in Water
Hydrophone(obscured)
1 m
Preamplifier electronics
48 cylinders
20
0
0.5
1
1.5
2
2.5
K
polymers
aluminum
copper brass
7x7 array of empty shells
15.4 cm
monopole
source
Titovich et al. JASA 2016
bulk modulus
circle to
square map
S
(ITC 1032)H
(NRL A48)
10 m
5.5 m
θ
Lens
Measurements made at Lake Travis Test Station of Applied Research Laboratories, UT Austin
Test Configuration
21
40 kHz
15 kHz
37.532.5
22.5
27.5
2520
35
30Lens of Shells
• Small # elements: 48
• Neutrally buoyant
• Broadband response
• Positive gain
Experimental Results
22Titovich et al. JASA 2016 doi: 10.1121/1.4948773
23
E.G. circle to rectangle mapping
Highly directional conformal TA
Map the omni-directional source (circle)
to one or two directions (polygon faces)
A AB B
25 kHz
45 kHz
65 kHz
35 kHz
55 kHz
75 kHz
circle to rectangle mapping
Highly directional TA
25
Highly directional TA
E.G. circle to triangle mapping
A
B
A
B
26
E.G. circle to triangle mapping
Highly directional TA
A
Fraction of energy in forward direction =
Circle to pair-of-arcs map
27
z-plane w-plane
a
b
47.5 kHz
Impedance matching
A
B
A
B
Impedance (and speed) can be matched - locally
CC
Impedance is matched at C
Impedance matching matched ray density
at C
Space-time modulation of a phononic crystalWillis effective equations
Outline
1 Space-time modulationDispersion relation - exact result
2 Willis equationsLow frequency limit of dispersion relation3D modulated elastic materialsTwo scale homogenization
Modulated Phononic Crystals: Non-reciprocal Wave Propagation
and Willis Materials. http://bit.do/modulated
2 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Equations in the moving frame
Space and time modulation
L1 L2
x = 0 x = γt0
t = 0
t = t0
3 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Wave propagation in 1D modulated phononic crystals
1D elastic mediumstress σ, particle velocity v , density ρ, elastic modulus κ
Stress-strain relation, momentum equation:
∂xv = ∂t(σ/κ), ∂xσ = ∂t(ρv), (1)
⇒ ∂xη + ∂t(Aη) = 0, (2)
with
η =
[
v
−σ
]
, A =
[
0 1/κρ 0
]
. (3)
Space and time modulation:
κ = κ(x , t) = κ(x − cmt), ρ = ρ(x , t) = ρ(x − cmt),
4 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Wave propagation in 1D modulated phononic crystals
1D elastic mediumstress σ, particle velocity v , density ρ, elastic modulus κ
Stress-strain relation, momentum equation:
∂xv = ∂t(σ/κ), ∂xσ = ∂t(ρv), (1)
⇒ ∂xη + ∂t(Aη) = 0, (2)
with
η =
[
v
−σ
]
, A =
[
0 1/κρ 0
]
. (3)
Space and time modulation:
κ = κ(x , t) = κ(x − cmt), ρ = ρ(x , t) = ρ(x − cmt),
4 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Wave propagation in 1D modulated phononic crystals
1D elastic mediumstress σ, particle velocity v , density ρ, elastic modulus κ
Stress-strain relation, momentum equation:
∂xv = ∂t(σ/κ), ∂xσ = ∂t(ρv), (1)
⇒ ∂xη + ∂t(Aη) = 0, (2)
with
η =
[
v
−σ
]
, A =
[
0 1/κρ 0
]
. (3)
Space and time modulation:
κ = κ(x , t) = κ(x − cmt), ρ = ρ(x , t) = ρ(x − cmt),
4 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Equations in the moving frame
Space and time modulation κ = κ(ξ), ρ = ρ(ξ), ξ = x − cmt
L1 L2
x = 0 x = γt0
t = 0
t = t0
Modulation speed cm
Periodic: L in space, L/cm in timePhase velocity and impedance: c =
√
κ/ρ, z =√
ρκ∀ ξassume c(ξ) > |cm| (subsonic) or c(ξ) < |cm|) (supersonic)η(x , t) → η(ξ, t)
∂xη + ∂t(Aη) = 0 ⇒ ∂ξ[(I − cmA)η] + A∂tη = 0 (4)
A depends on ξ but not on t5 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Change of state vector / Dispersion relation
η → ψ = (I − cmA)η ⇒
∂ξψ + B(ξ)∂tψ = 0 (5)
where B = A(I − cmA)−1 =1
c2 − c2m
[
cm 1/ρκ cm
]
Floquet-Bloch solutions
ψ(ξ, t) = ψ(ξ)e−iΩt , ψ(ξ + L) = ψ(ξ)eiKL, (6)
wavenumber K , frequency Ω in the moving frame (ξ, t)• In fixed frame: kx − ωt = Kξ − Ωt ⇒ k = K , ω = Ω+ cmk
6 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Dispersion relation
∂ξψ+B(ξ)∂tψ = 0, ψ(ξ, t) = ψ(ξ)e−iΩt , ψ(ξ+L) = ψ(ξ)eiKL
Dispersion relation: det(
M(ξ + L, ξ) − eiKLI)
= 0
Matricant ∂ξM = iΩB(ξ)M, M(ξ0, ξ0) = I
Form of B = cm
c2−c2
m
I + c
c2−c2
m
J, J =
[
0 1/z
z 0
]
⇒ cos
(
KL −⟨
cmΩL
c2 − c2m
⟩)
=1
2tr N(ξ + L, ξ)
Unitary matrix N: ∂ξN = iΩc
c2−c2
m
JN, N(ξ0, ξ0) = I
N equation that of a spatially modulated medium, wave speed
c(x) − c2m
c(x) ⇔ standard Bloch-Floquet for a periodic medium
7 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Dispersion relation
∂ξψ+B(ξ)∂tψ = 0, ψ(ξ, t) = ψ(ξ)e−iΩt , ψ(ξ+L) = ψ(ξ)eiKL
Dispersion relation: det(
M(ξ + L, ξ) − eiKLI)
= 0
Matricant ∂ξM = iΩB(ξ)M, M(ξ0, ξ0) = I
Form of B = cm
c2−c2
m
I + c
c2−c2
m
J, J =
[
0 1/z
z 0
]
⇒ cos
(
KL −⟨
cmΩL
c2 − c2m
⟩)
=1
2tr N(ξ + L, ξ)
Unitary matrix N: ∂ξN = iΩc
c2−c2
m
JN, N(ξ0, ξ0) = I
N equation that of a spatially modulated medium, wave speed
c(x) − c2m
c(x) ⇔ standard Bloch-Floquet for a periodic medium
7 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Dispersion relation
∂ξψ+B(ξ)∂tψ = 0, ψ(ξ, t) = ψ(ξ)e−iΩt , ψ(ξ+L) = ψ(ξ)eiKL
Dispersion relation: det(
M(ξ + L, ξ) − eiKLI)
= 0
Matricant ∂ξM = iΩB(ξ)M, M(ξ0, ξ0) = I
Form of B = cm
c2−c2
m
I + c
c2−c2
m
J, J =
[
0 1/z
z 0
]
⇒ cos
(
KL −⟨
cmΩL
c2 − c2m
⟩)
=1
2tr N(ξ + L, ξ)
Unitary matrix N: ∂ξN = iΩc
c2−c2
m
JN, N(ξ0, ξ0) = I
N equation that of a spatially modulated medium, wave speed
c(x) − c2m
c(x) ⇔ standard Bloch-Floquet for a periodic medium
7 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Dispersion relation / Main result
Space and time modulation c = c(ξ), x = x(ξ), ξ = x − cmt
L1 L2
x = 0 x = γt0
t = 0
t = t0
C (cm, c, z) = dispersion curve of the modulated laminate
(k, ω) ∈ C (cm, c, z) ⇐⇒ T (k, ω) ∈ C (0, c − c2m
c, z)
with exact transformation
T (k, ω) =
(
k − cm(ω − cmk)
⟨
1
c2 − c2m
⟩
, ω − cmk
)
8 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Dispersion relation / Example: bilaminate
L1 L2
x = 0 x = γt0
t = 0
t = t0
cm = 0, spatially modulated
cos kL = cos ωc1
L1 cos ωc2
L2 − 1
2
(
z1
z2+ z2
z1
)
sin ωc1
L1 sin ωc2
L2
cm 6= 0, space-time modulation
cos
[
kL − cm(ω − cmk)
(
L1
c2
1− c2
m
+L2
c2
2− c2
m
)]
= cos
(
ω − cmk
c2
1− c2
m
c1L1
)
cos
(
ω − cmk
c2
2− c2
m
c2L2
)
−
1
2
(
z1
z2
+z2
z1
)
sin
(
ω − cmk
c2
1− c2
m
c1L1
)
sin
(
ω − cmk
c2
2− c2
m
c2L2
)
9 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Dispersion relation / Example: bilaminate
(k, ω) → T (k, ω) induces a shearing effect at small speed cm = γ.Introduces assymetric band gaps.
γ = 0 γ = 0.1 min ϕ γ = 0.8 min ϕ
γ = 4 max ϕ γ = 1.52 max ϕ γ = 1.16 max ϕ
Figure : Transient responses for acentral frequency f1 (a − c) andf2 (d − f ). Green arrows cm = 0;blue arrows cm 6= 0. Propagationis symmetric for cm = 0 (b, e).For cm 6= 0, right-going waves areaccelerated (c, f ) and left-goingwaves are blocked at f1 (c) anddecelerated at f2 (f ).
10 / 19
Space-time modulation of a phononic crystalWillis effective equations
Governing equationsEquations in the moving frameDispersion relation
Willis equations in the low frequency limit / Example
(a)
(b)
(c)
(d)
(e)
(f)
Figure : Transient responses for acentral frequency f1 (a − c) andf2 (d − f ). Green arrows cm = 0;blue arrows cm 6= 0. Propagationis symmetric for cm = 0 (b, e).For cm 6= 0, right-going waves areaccelerated (c, f ) and left-goingwaves are blocked at f1 (c) anddecelerated at f2 (f ).
11 / 19
Space-time modulation of a phononic crystalWillis effective equations
Low frequency limit of dispersion relation3D modulated elastic materialsTwo scale homogenization
Willis equations in the low frequency limit
The low frequency limit ⇔ a Willis-type equation:
κe∂2
x U + 2s∂x∂tU − ρe∂2
t U = 0
where κe=
⟨
κ
κ − c2mρ
⟩
2⟨
1
κ − c2mρ
⟩
−1
− c2
m
⟨
ρκ
κ − c2mρ
⟩
,
ρe= −c
2
m
⟨
ρ
κ − c2mρ
⟩
2⟨
1
κ − c2mρ
⟩
−1
+
⟨
ρκ
κ − c2mρ
⟩
,
s = cm
⟨
κ
κ − c2mρ
⟩⟨
ρ
κ − c2mρ
⟩⟨
1
κ − c2mρ
⟩
−1
− cm
⟨
ρκ
κ − c2mρ
⟩
.
Non-reciprocal non-dispersive phase speeds:
V = cm +
(
⟨
cm
c2 − c2m
⟩
±√
⟨
c/z
c2 − c2m
⟩⟨
cz
c2 − c2m
⟩
)
−1
,
12 / 19
Space-time modulation of a phononic crystalWillis effective equations
Low frequency limit of dispersion relation3D modulated elastic materialsTwo scale homogenization
Willis equations in the low frequency limit
The low frequency limit ⇔ a Willis-type equation:
κe∂2
x U + 2s∂x∂tU − ρe∂2
t U = 0
kLπ−π
ω/ωc
1
C (γ)C (0)
Non-reciprocal non-dispersive phase speeds:
V = cm +
(
⟨
cm
c2 − c2m
⟩
±√
⟨
c/z
c2 − c2m
⟩⟨
cz
c2 − c2m
⟩
)
−1
,
13 / 19
Space-time modulation of a phononic crystalWillis effective equations
Low frequency limit of dispersion relation3D modulated elastic materialsTwo scale homogenization
3D modulated elastic materials
n
m Figure : Sketch of a 3Dtwo-phase laminate.Modulation direction n .Direction m is a genericorthogonal direction.
Governing equations
Equilibrium ∇ · σ = ∂tp
Constitutive, stress, momentum: σ = C : ε, p = ρv
Strain, displacement, velocity: ε = ∇⊗s u, v = ∂tu
Space-time modulation
∇ · [C(x − cmt) : (∇⊗s u(x, t))] = ∂t(ρ(x − cmt)∂tu(x, t)).14 / 19
Space-time modulation of a phononic crystalWillis effective equations
Low frequency limit of dispersion relation3D modulated elastic materialsTwo scale homogenization
Two scale homogenization of modulated 3D material
u(x, t) → u(x, ζ, t) = U(x, ξ, t), where ξ = x − cmt, ζ = ξǫ
Results for the the homogenization limit ǫ → 0
Equilibrium ∇ · Σ = ∂tP
Coupled stress, momentum constitutive relations of Willis form:
Σ = Ce : E + S1 · V ,
P = S2 : E + ρe · V ,
with symmetries Ce = (Ce)T , ρe = (ρe)T , S1 = −(S2)T
Strain, displacement, velocity:E = E(x, t) = ∇⊗s U(x, t), V = V (x, t) = ∂tU(x, t)
15 / 19
Space-time modulation of a phononic crystalWillis effective equations
Low frequency limit of dispersion relation3D modulated elastic materialsTwo scale homogenization
Effective parameters / dispersion relation
Willis parameters
Ce = 〈C〉 + 〈C : n⊗Γ〉 · 〈Γ〉−1 · 〈Γ⊗n : C〉 − 〈C : n⊗Γ⊗n : C〉 ,
S1 = cm 〈C : n⊗Γ〉 · 〈Γ〉−1 · 〈ρΓ〉 − cm 〈ρC : n⊗Γ〉 ,
S2 = −cm 〈ρΓ〉 · 〈Γ〉−1 · 〈Γ⊗n : C〉 + cm 〈ρΓ⊗n : C〉 ,
ρe = 〈ρ〉 I − c2
m 〈ρΓ〉 · 〈Γ〉−1 · 〈ρΓ〉 + c2
m
⟨
ρ2Γ⟩
where Γ = (n · C · n − c2mρI)−1
Macroscopic dispersion relation:
det(
k · Ce · k − 2ωk · S − ω2ρe)
= 0
16 / 19
Space-time modulation of a phononic crystalWillis effective equations
Low frequency limit of dispersion relation3D modulated elastic materialsTwo scale homogenization
Dispersive system
non-modulated modulated non-modulated
Incident wave (ω, q).
17 / 19
29
• Transformation Acoustics
Nonuniqueness
Pentamode materials one wave property
- Negative index, GRIN lenses
- Anisotropy not yet exploited
- Conformal TA - Highly directional lensing
• Space-time modulated (activated) phononic crystal
Exact result for wave-like modulation
Low frequency limit: Willis equations
Non-reciprocal in wave speeds
Non-reciprocal coupling in transmission/reflection
& to you
for listening!
ONR NSF CNRS CIES
Alexey Titovich Rutgers
Adam Nagy ..
Xiaoshi Su ..
Jeffrey Cipolla Weidlinger
A.C. Hladky-Hennion Lille/IEMN
M.R. Haberman ARL/UT
C. Cushing ..
Hussein Nassar U. Missouri
Guoliang Huang ..
Thanks
30