Transformation Acoustics, Pentamode Lenses and …...Andrew Norris Rutgers University NJ, USA Transformation Acoustics, Pentamode Lenses and Spece-Time Modulated Phononic Crystals

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


E.G. circle to triangle mapping

A

B

A

B

26

E.G. circle to triangle mapping


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

http://bit.do/modulated


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



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






∂xv = ∂t(σ/κ), ∂xσ = ∂t(ρv), (1)

⇒ ∂xη + ∂t(Aη) = 0, (2)

with

η =

[

v

−σ

]

, A =

[

0 1/κρ 0

]

. (3)



4 / 19






∂xv = ∂t(σ/κ), ∂xσ = ∂t(ρv), (1)

⇒ ∂xη + ∂t(Aη) = 0, (2)

with

η =

[

v

−σ

]

, A =

[

0 1/κρ 0

]

. (3)



4 / 19



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



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



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



Dispersion relation




= 0


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)


c2−c2

m

JN, N(ξ0, ξ0) = I


c(x) − c2m


7 / 19



Dispersion relation




= 0


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)


c2−c2

m

JN, N(ξ0, ξ0) = I


c(x) − c2m


7 / 19



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



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



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



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


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



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



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



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



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



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

Documents

Transformation Acoustics, Pentamode Lenses and …...Andrew Norris Rutgers University NJ, USA Transformation Acoustics, Pentamode Lenses and Spece-Time Modulated Phononic Crystals