Introductory lecture on nonlinear acoustics (PDF)

Introductory Lecture on Nonlinear Acoustics

Laboratoire d’Acoustique de l’Université du Maine (LAUM)UMR-CNRS 6613, Le Mans, France

Vincent TOURNATResearch Scientist at CNRS

[email protected]

Oléron 02/06/2014

mailto:[email protected]

Main goals of the lecture

➪ A basic culture on nonlinear acoustics for researchers

➪ Basic theory : derivations of the main equations

➪ Overview of some nonlinear effects and applications

➪ Basic theory : some resolution methods, some solutions

21. Introduction PageV. Tournat - Introductory lecture on nonlinear acoustics 2

Main sources for this lecture

Introductory lectures of the previous summer schools on nonlinear acoustics O. Bou Matar (fluids 2007), C. Barrière (solids 2007), F. Coulouvrat (Fluids & Solids 2010)

Textbooks Nonlinear acoustics, M.F. Hamilton, D.T. Blackstock Ed., Academic Press, AIP (2008). Theoretical foundations of nonlinear acoustics, O.V. Rudenko and S.I. Soluyan, Studies in Soviet Science - Physical Sciences (1977). Nonlinear underwater acoustics, B.K. Novikov, O.V. Rudenko, V.I. Timoshenko, ASA, AIP (1987).

1. Introduction

2. Constitutive equations of nonlinear acoustics in fluids

3. Burgers equation, resolution methods, solutionsApproximated solutions, shock formation, harmonic generation…

Towards Westervelt equation

4. KZK equation, generalized Burgers equations, and related problems

Fluids

6. Analysis of the nonlinear propagation equation in isotropic solids

7. Nonclassical nonlinearities in solids

5. Constitutive equations of nonlinear acoustics in solidsLagrangian description, specificities of solids, similarities with fluids

Solid

s


Case study of the parametric emitting antenna


Various fundamental effects and examples

The «system» approach of nonlinearity

LinearInput Output

e1(t) = E1 sin(!1t) s1(t) = S1 sin(!1t+ �1)

e2(t) = E2 sin(!2t) s2(t) = S2 sin(!2t+ �2)

e1(t) = E1 sin(!1t) s1(t) = S1 sin(!1t+ �1)

↵e1(t) + �e2(t) ↵s1(t) + �s2(t)

Superposition principle

Nonlinear↵e1(t) + �e2(t) 6= ↵s1(t) + �s2(t)

Amplitude dependent effects New frequencies ...


Another view of nonlinearity

Input, excitation

Output, response

Linear systemSuperposition principle applies

(approximation)

➪ Linearity is an approximation (sometimes good) of any intrinsically nonlinear system

Input, excitation

Output, response

Nonlinear systemAny real systems


Sounds

10�1 100 101 102 103

Frequency (Hz)

104 105 106 107 108 109 1010 1011

Infrasounds

Phenomenas over a wide frequency range

Ultrasounds Hypersounds


Shock wave in a trombone


A. Hirschberg, J. Gilbert, R. Msallam, A.P.J. Wijnands, J. Acoust. Soc. Am. 99, 1754-1758 (1996)

Microphone 1, tube entrance

Microphone 2, at a distance of several

wavelengths from the tube entrance

Medical imaging with harmonics

http://www.msdlatinamerica.com/ebooks/CoreCurriculumTheUltrasound/sid102252.html#F25-1

➪ A better contrast (scattering and first reflexions are lowered)➪ A better resolution (smaller wavelength)➪ Less effects of spurious sidelobes


Image at the emitted frequency Image at twice the emitted frequency

http://www.msdlatinamerica.com/ebooks/CoreCurriculumTheUltrasound/sid102252.html#F25-1

!1

!2

⌦ = !1 � !2

Parametric emitting antenna (difference frequency generation / self-demodulation)

Emitter

➪ Underwater acoustics in the 1950s

➪ Long range and high directivity


Parametric receiving antenna

!

⌦

! ± ⌦

Emitter Receiver


DC effects, streaming, levitation radiation pressure,…

Particle trapping, segregation, levitation Fluid flow, micro-fluidics


➪ CR7 : Microfluidique (M. Baudouin)

➪ CR1 : Pression de radiation et interfaces (R.Wunenberger)

➪ CR10 : Thermo-acoustique (G. Penelet)

Thermal expansion in solids: nonlinear phonons

Distance

Potential

Inter-atomic potential

Low amplitude phonons(thermal agitation)

Harmonic potential

Distance

Potential

Inter-atomic potential

Larger amplitude phonons(Larger T)

Anharmonic potential

Shifted average position = dilatation


Introductory Lecture on Nonlinear AcousticsFlu

ids1. Introduction








Solid

s



14

Constitutive equations up to the second order in acoustic quantities

⇢ = ⇢0 + ⇢a

~v = ~v0 + ~va

P = P0 + pa

Acoustic quantities are usually very small compared to the static ones

Mach number

2. Constitutive equations PageV. Tournat - Introductory lecture on nonlinear acoustics 14

➪ Second order equations are most of the time sufficient for nonlinear acoustics in fluids

(154 dB in air ➪ M~0.01)

Equation of motion (momentum conservation)

⇢D~v

Dt= �~rP

⇢ = ⇢0 + ⇢a ~v = ~v0 + ~va P = P0 + pa

1st order 2nd order

⇢0@~va@t

+ ~rpa = �⇢a@~va@t

� ⇢0(~va.~r)~va


D

Dt=

@

@t+ ~v.~r Total (material) derivative over time

Convective term, Eulerian description

L = Ec � Ep =1

2⇢0~v

2a �

p2a2⇢0c20

Introducing Lagrangian energy density


1st order 2nd order

⇢0@~va@t

+ ~rpa = �⇢a@~va@t

� ⇢0(~va.~r)~va

Using plane wave relations

⇢a@~va@t

= �~r✓

p2a2⇢0c20

◆

⇢0@~va@t

+ ~rpa = �~rL ➪ Equation of motion

Mass conservation law

@⇢

@t+ ~r.(⇢~v) = 0

@⇢a@t

+ ⇢0~r.~va = �⇢a~r.~va � ~va.~r⇢a

1st order 2nd order

(exact development)


⇢ = ⇢0 + ⇢a ~v = ~v0 + ~va


Equation of state, fluid comportment law

Taylor expansion of the pressure-density relationship

A = ⇢0

✓@P

@⇢

◆

⇢0,s

= ⇢0c20 Inverse of compressibility coefficient

Variation of the sound celerity due to a surpression

B = ⇢20

✓@2P

@⇢2

◆

⇢0,s

= 2⇢20c30

✓@c

@p

◆

⇢0,s

@⇢a@t

+ ⇢0~r.~va = �⇢a~r.~va � ~va.~r⇢a


⇢0@~va@t

+ ~rpa = �~rL ➪ Equation of motion

Constitutive equations up to the second order

➪ Continuity equation

➪ Equation of state

20

Navier-Stokes equation (momentum conservation)

⇢D~v

Dt= ~r(�P I+ ⌧ )


Viscous stress tensor

Strain rate tensor

⇢0@~va@t

+ ~rpa �✓⇣ +

4

3⌘

◆~�~va = �~rL

Equation of state, fluid comportment law

P = P0 +

✓@P

@⇢

◆

⇢0,s

⇢a +1

2

✓@2P

@⇢2

◆

⇢0,s

⇢2a +

✓@P

@s

◆

⇢,s0

sa +O(⇢3a, s2a)

A Taylor expansion around values at rest ⇢0 s0


⇢a ' pac20

� B

2A

p2a⇢0c40

+

⇢0c40

✓1

CV� 1

CP

◆@pa@t

After resolution of the heat equation, acoustic temperature propagation equation,

Specific heat at constant volume and pressure

Thermal conductivity

@⇢a@t

+ ⇢0~r.~va = �⇢a~r.~va � ~va.~r⇢a


➪ Equation of motion

Constitutive equations up to the second order

➪ Continuity equation

➪ Equation of state

⇢0@~va@t

+ ~rpa �✓⇣ +

4

3⌘

◆~�~va = �~rL

⇢a ' pac20

� B

2A

p2a⇢0c40

+

⇢0c40

✓1

CV� 1

CP

◆@pa@t

⇢0@~va@t

+ ~rpa �✓⇣ +

4

3⌘

◆~�~va = �~rL

⇢a ' pac20

� B

2A

p2a⇢0c40

+

⇢0c40

✓1

CV� 1

CP

◆@pa@t

@⇢a@t

+ ⇢0~r.~va = �⇢a~r.~va � ~va.~r⇢a


Derivation of the nonlinear propagation equation

@

@t

Nonlinear propagation equation

�pa �1

c20

@2pa@t2

+b

c20

@3pa@t3

= � �

⇢0c40

@2p2a@t2

�✓�+

1

c20

@2

@t2

◆L

� = 1 +B

2A➪ Parameter of quadratic nonlinearity

From convection From material behavior


b =1

⇢0c20

✓⇣ +

4

3⌘

◆+

⇢0c20

✓1

CV� 1

CP

◆

Dissipation (just an imaginary part in the wave number for monochromatic waves)

Attenuation coefficient

↵(!) =b!2

2c0=

1

à

Acoustic diffusivity

Parameter of quadratic nonlinearity

� = 1 +B

2A

From convection From material

Medium B/A

Distilled water (20°C) 5.0

Methanol (20°C) 9.6

Monoatomic gas (20°C) 0.67

Biological media 5-12 (typ.)


Simplification of the nonlinear propagation equation

�pa �1

c20

@2pa@t2

+b

c20

@3pa@t3

= � �

⇢0c40

@2p2a@t2

�✓�+

1

c20

@2

@t2

◆L

For plane waves, at the first order: va =pa⇢0c0

2 reasons to neglect this term

L = 0➪

The effect is local and not cumulative


Westervelt equation (1963)

�pa �1

c20

@2pa@t2

+b

c20

@3pa@t3

= � �

⇢0c40

@2p2a@t2

(PJ Westervelt 1919- )

273. Model equations PageV. Tournat - Introductory lecture on nonlinear acoustics


3. Model equations PageV. Tournat - Introductory lecture on nonlinear acoustics 28

Fluids

1. Introduction




4. KZK equation, generalized Burgers equations, and related problems Case study of the parametric emitting antenna

�pa �1

c20

@2pa@t2

+b

c20

@3pa@t3

= � �

⇢0c40

@2p2a@t2

Westervelt equation

✓@

@z� 1

c0

@

@t+

b

2c0

@2

@t2+

�pa⇢0c30

@

@t

◆✓@

@z+

1

c0

@

@t� b

2c0

@2

@t2� �pa

⇢0c30

@

@t

◆pa = 0

Backward propagation Forward propagation

In one dimension (plane waves), factorizing the operators,

Simplifications of Westervelt equation


➪ One way approximation

Burgers equation (1948)

@pa@z

+1

c0

@pa@t

� b

2c0

@2pa@t2

� �pa⇢0c30

@pa@t

= 0

@pa@z

+1

c

@pa@t

� b

2c0

@2pa@t2

= 0

Factorizing these two terms, we

obtain

c = c0

✓1 +

�pa⇢0c20

◆with an amplitude dependent wave velocity:


31

Shock formation, harmonic generation


32


-0.5 0 0.5-1

-0.5

0

0.5

1

o

p a

Positive part arrives earlier

Negative part arrives later


c = c0

✓1 +

�pa⇢0c20

◆

⌧ = t� z

c0z0 = µz

Method of the slowly varying profile

The profile changes slowly in spaceµ ⌧ 1

The coordinate system moves with the wave

@

@z= � 1

c0

@

@t+ µ

@

@z0@

@⌧=

@

@t

@pa@z0

� b

2c0

@2pa@⌧2

� �pa⇢0c30

@pa@⌧

= 0


Normalizing the equation variables by: ⇠ =z0

`nl✓ = !⌧ P =

pap0

pa = p0 sin(!⌧)Considering a monochromatic wave excitation of the form:

With the characteristic nonlinear length: `nl =⇢0c30!�p0

=1

k0�M

The characteristic nonlinear length


Normalization

Normalized Burgers’ equation

`nl =⇢0c30!�p0

=1

k0�M� =

à`nl

=b!k20�M

2=

bk30�p02⇢0c0

Gol’dberg number

à =2c0b!2


Some few characteristic numbers

Sonic boom in air: pa ' 100 Pa

pa ' 5 MPaClose to a jet engine:

Lithotripsy (water): pa ' 108 Pa

M ' 7.10�4

M ' 5.10�2

M ' 5.10�2 `nl ' 0.15 m

`nl ' 60 m

`nl < 1 m


Solving Burgers’ equation

Without dissipation:@pa@z

+1

c0

@pa@t

� �pa⇢0c30

@pa@t

= 0

@pa@z

+1

c

@pa@t

= 0


Monochromatic source

➪ Implicite solution (Poisson’s solution, exact)

c = c0

✓1 +

�pa⇢0c20

◆


pa(z, ⌧) = p0 sin

✓!

✓⌧ +

z

c0

�M

1 + �M

◆◆

This implicit function takes non unique values for ⇠ =z

`nl> 1





Several ways to describe the profile distortion, shock formation and shape evolution

➪ The Fubini solution

➪ The Rankine-Hugoniot relations➪ The rule of equal surfaces (Landau)

➪ The Hopf-Cole transformation (exact solution)

➪ The Fay solution (old age solution)


➪ The Poisson solution

➪ The quasi-linear approximation (successive approximations)

➪ The Burgers-Hayes method

➪ …

➪ The Hopf-Cole transformation

The adequate variable change

➪ A 1D linear diffusion equation

➪ Many well-known solutions

Solving Burgers equation: exact solution


(E. Hopf, 1950)(J.D. Cole, 1951)

0 5 10 15 200

0.2

0.4

0.6

0.8

1

➪ The Hopf-Cole transformation(E. Hopf,1950)(J.D. Cole, 1951)

Monochromatic source (J.S. Mendousse, 1953)

Solving Burgers’ equation: exact solution

Modified Bessel functions


0 5 10 15 200

0.2

0.4

0.6

0.8

1




(Cole 1951, Mendousse 1953)


-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1



Shock formation distance Maximum shock amplitude

Sawtooth shape Frequency dependent attenuation Old age profile

Initial profile

Pre-shock region Transition region

Sawtooth region

Solving Burgers’ equation: Fubini solutionHarmonic balance method + slowly varying profile

We search solutions in the form of a sum of harmonic functions (a Fourier serie):

pa = p0

+1X

n=1

Bn(⇠) sin(n!⌧)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

⇠

Bn


Bn(⇠) =2Jn(n⇠)

n⇠

(Fubini 1935)

Solving Burgers’ equation: quasi-linear approximation

The successive approximation method

pa(z, ⌧) = µp(1)a (z, ⌧) + µ2p(2)a (z, ⌧)


(weak waves, second harmonic generation)

At the first order in µ

At the second order in µ

➪ 2 linear problems

Nonlinear source term

(Gol’dberg 1957)



Primary wave / pump wave

(Gol’dberg 1957)

Using the boundary condition at

General solution Particular solution

Secondary wave


➪ Could be continued, third harmonic, cascade process


(Gol’dberg 1957)

↵z ⌧ 1Close enough to the source

➪ Fubini solution for

➪ Accurate approximation for or and

Solving Burgers’ equation: Fay solution


⇠ � � � 1

(strong waves, harmonic generation, old age) (Fay 1931)

➪ Independent of the excitation amplitude !

➪ Nonlinear saturation mechanism

(exact solution)

Asymptotic expansion in the case

Solving Burgers’ equation: N waves


-0.5 0 0.5-1

-0.5

0

0.5

1

➪ Profile shape typical of supersonic objects (sonic boom), electric sparks, bursting balloons…

A pulse starting with a compression

➪ CR11 : Ondes de choc Atm. (F. Coulouvrat)

52

Introductory Lecture on Nonlinear Acoustics1. Introduction





Fluids


Diffraction Dissipation Nonlinearity

Parabolic equation of diffraction

Other model equations: the KZK equation

z0 = µz

x

0 =pµx

y0 =pµy

⌧ = t� z

c0

➪ The coordinate system is changed for this one, in the 3D Westervelt equation

@2pa@z0@⌧

=c02�?pa +

b

2c0

@3pa@⌧3

+�

2⇢0c30

@2p2a@⌧2

4. KZK, Generalized Burgers… PageV. Tournat - Introductory lecture on nonlinear acoustics 53

KZK equation

@2pa@z0@⌧

=c02�?pa +

b

2c0

@3pa@⌧3

+�

2⇢0c30

@2p2a@⌧2

➪ A very important equation for beam problems in nonlinear acoustics and for numerics

Zabolotskaya & Khokhlov 1969, Kuznetsov 1971

`d =k0a2

2

N =`nl`d

Khokhlov number

PageV. Tournat - Introductory lecture on nonlinear acoustics 54

Near fieldFresnel zone

Far fieldFraunhofer zone

Rayleigh distance

Characteristic diffraction length

Focal distance

Focal zone

4. KZK, Generalized Burgers…

Solutions of KZK equation


➪ Successive approximation method depending on the orders of magnitude

➪ Decomposition of the source pattern in a sum of Gaussian functions

➪ Simple Gaussian source or piston

� =à`nl

=b!k20�M

2=

bk30�p02⇢0c0

N =`nl`d




➪ Second harmonic generation, Gaussian beam, quasi-linear approximation

Source signal

First order equation

Primary wave solution(Green’s function method)

Second order equation




Primary wave solution(Green’s function method)

Second harmonic wave solution(Green’s function method)

For vanishing attenuation

➪ A beam narrower by a factor

➪ A maximum efficiency on the z-axis


Example of the emitting parametric antenna

!1

!2

⌦ = !1 � !2

Emitter


Firstly used in underwater acoustics from the 1950s

➪ Low frequency sound (low attenuation) radiated with a high directivity using

indirectly a relatively small size emitter

Virtual sources for the LF sound


➪ In 1D, the solutions are straightforward (successive approximations, Burgers’ equation)

At the first order in µ

At the second order in µ


Nonlinear source term

Difference frequency generation



➪ For wave packets or arbitrary modulation function

P�(z, ⌧) =�p202b!2

(1� e�2z/à)@f2(⌦⌧)

@⌧

f(⌦⌧)

➪ The wave profile is the derivative of the pump intensity envelope



Self-demodulation




M.A. Averkiou, Y.S. Lee, M.F. Hamilton, Self-demodulation of amplitude and frequency modulated pulses in a thermoviscous fluid, J. Acoust. Soc. Am 94, 2876-2883 (1993).


➪ For diffracting beams (KZK equation with successive approximation method and Gaussian beams)

Westervelt regime

Berktay regime

Generation in the pump wave near field, before diffraction

`nl < à < `d

Generation by diffracted pump waves

`d < `nl < à



Pump waves Demodulated waves



Parametric antenna: beam width


Westervelt regime

`nl < à < `d

LF wavelength

Effective length of the NL sources, limited by absorption

Berktay regime

`d < `nl < àDiffraction length of the pump waves (~effective length of the NL sources)

(at small angles)

➪ The directivity depends on the effective length of the NL virtual sources


Parametric antenna in air


http://www.lradx.com/site/


Parametric antenna in air: audible frequency range




Generalized Burgers’ equations

➪ For cylindrical and spherical waves

@pa@r

+m

rpa �

b

2c0

@2pa@⌧2

� �pa⇢0c30

@pa@⌧

= 0

For cylindrical wavesFor spherical waves

➪ Also for horns and tubes with slowly varying section

m = 1 m = 1/2

@qa@z

=�qa⇢0c30

@qa@⌧

qa = (r/r0)pa qa = (r/r0)1/2pa

z = ±r0 ln(r/r0) z = ±2(pr �

pr0)

pr0

PageV. Tournat - Introductory lecture on nonlinear acoustics 664. KZK, Generalized Burgers…

Converging cylindrical shock waves in liquids


T. Pezeril et al., Direct Visualization of Laser-Driven Focusing Shock Waves, Phys. Rev. Lett. 106, 214503 (2011).


Generalized Burgers’ equations➪ For different types of linear or nonlinear processes

@pa@z

=�pa⇢0c30

@pa@⌧

+ L(pa)


L(pa) /@3pa@⌧3

Pure dispersion ➪ Korteweg-de-Vries equation

N. Sugimoto et al., Experimental demonstration of generation and propagation of acoustic solitary waves in an air-filled tube. Phys. Rev. Lett. (1999) vol. 83 (20) pp. 4053


Generalized Burgers’ equations➪ For different types of linear or nonlinear processes

@pa@z

=�pa⇢0c30

@pa@⌧

+ L(pa)

L(pa) /@3pa@⌧3

Pure dispersion ➪ Korteweg-de-Vries equation

L(pa) /@2p2a@⌧2

Nonlinear dissipation

L(pa) =m

2c0

@

@⌧

Z ⌧

�1e�(⌧�⌧ 0)/⌧R @pa(z, ⌧ 0)

@⌧ 0d⌧ 0 Relaxation process

L(pa) = �b

r2

⇡

Z ⌧

�1

@pa(z, ⌧ 0)

@⌧ 0d⌧ 0p⌧ � ⌧ 0

Viscous and thermal losses at the walls of a tube

L(pa) = ↵@n

@⌧n

Z ⌧

�1

@pa(z, ⌧ 0)

@⌧ 0d⌧ 0

(⌧ � ⌧ 0)�Complex visco-elastic media, biological media


Distributed physical and geometrical vs

Localized nonlinearities


Side holes in tubes (nonlinear dissipation by vortices)Helmholtz resonatorsHoles in plates, porous materials (Forcheimer’s nonlinearity)Gaz bubbles, soft shells in liquids


Mach number

Characteristic nonlinear length

Attenuation length

Diffraction length

Khokhlov number

Gol’dberg number

Summary of the characteristic parameters

`nl =⇢0c30!�p0

=1

k0�M

à =2c0b!2

M =vac0

=pa⇢0c20

� =à`nl

=b!k20�M

2=

bk30�p02⇢0c0

N =`nl`d

`d =k0a2

2


Summary


Constitutive equations of nonlinear acoustics in fluids

One-way and 1D approximation of Westervelt equation ➪ Burgers equation

Shock formation, harmonic generation…

Second order in acoustic quantities, cumulative nonlinear terms

KZK equationCase study of the parametric emitting antenna

➪ Westervelt equation

Some resolution methods, solutions

Generalized Burgers equations

Nonlinear beams, diffraction

Various processes, geometries

Various effects (frequency domain)

f

A

Harmonic cascade f

A

Nonlinear modulationSuppression of sound by sound

f

A

Self-demodulation

f

A

DC effects, streaming f

A

Subharmonic generation f

A

Transfer of modulation

73PageV. Tournat - Introductory lecture on nonlinear acoustics 73

And many other processes, not presented


Reflexion of shock waves, by shock wavesNonlinear coupling with viscous and thermal effects (boundary layers)

Parametric reception

Dynamic / temporal interactions

Standing waves, interaction of counter-propagating waves

Inhomogeneous media, bubbly media

Parametric amplification

Phase singularities…

Statistical phenomena, interactions with noise

➪ CR5 : Aéroacoustique (C. Bailly) ➪ CR9 : Bulles, mousses (V. Leroy)




[email protected]

Oléron 02/06/2014







Fluids




Solid

s



Solids Fluids➪ Support static shear stress, shear waves ➪ Do not support static shear stress,

and most of the time shear waves➪ No large particle displacement

➪ Fluid particles can go far away

➪ Exhibit Anisotropy

➪ The solid has a particular shape at rest

➪ Liquids take the shape of the container


Lagrangian description

Eulerian description

➪ We follow the material particle displacement relative to its position at rest

➪ We observe the medium change at a given fixed geometrical position

Mt

x1

x2

O

M0

~x

f(~x, t)

Df

Dt=

@f

@t+ ~v.~rf

~a

x1

x2

O

M0

~x

Mt

~U

~

U = ~x� ~a

~x = ~x(~a, t)


➪ For nonlinear acoustics in solids the Lagrangian description is usual

x

aU = x� a

Natural (equilibrium)

position

Current coordinate of a particle

Lagrange displacement (displacement relative to the

natural position)

~v =@U

@t=

DU

Dt

➪ The total (particular) derivative is equal to the partial derivative


Deformation gradient tensor~a

x1

x2

O

~x

d~a

d~x

Green strain tensor (Lagrangian)

E =1

2(FT.F� I) Eij =

1

2

✓@Ui

@aj+

@Uj

@ai+

@Uk

@ai

@Uk

@aj

◆

F =@x

@a=

@(a+U)

@a= I+

@U

@a


The mass conservation law

dV = (detF)dV0

⇢dV = ⇢0dV0

⇢

⇢0=

1

detF


Deformed state State at rest

Harmonic potential Nonlinear term(linear elastic behavior) (quadratic elastic behavior)

⇢0W =1

2!CijklEijEkl +

1

3!CijklmnEijEklEmn + . . .

Taylor expansion of the elastic strain energy

The material behavior : elastic energy as a function of strain


Second order elastic coefficients

Third order elastic coefficients

21 at most 56 at most

Cijkl Cijklmn

⇢0@2U

@t2= ra.P

Piola-Kirchhoff stress tensor (Lagrangian)

P =⇢0⇢�.(F�1)T = ⇢0F.

@W

@E

Equation of motion in Lagrangian coordinates

Pij = Cijkl@Uk

@al+

1

2Mijklmn

@Uk

@al

@Um

@an+

1

3Mijklmnpq

@Uk

@al

@Um

@an

@Up

@aq

Quadratic nonlinearity Cubic nonlinearity

Mijklmn = Cijklmn + Cijln�km + Cjnkl�im + Cjlmn�ik

Material nonlinearity Geometric nonlinearity (linear elastic constants)


⇢0@2Ui

@t2=

@Pij

@aj

⇢0@2Ui

@t2=

@2Uk

@aj@al

✓Cijkl +Mijklmn

@Um

@an

◆

Propagation equation in the general case


Mijklmn = Cijklmn + Cijln�km + Cjnkl�im + Cjlmn�ik

Material nonlinearity Geometric nonlinearity (linear elastic constants)

5. Constitutive equations

➪ Voigt’s notation can be usedCijklmn

Cijkl C↵�

C↵��

0

@A11 A12 A13

A21 A22 A23

A31 A32 A33

1

A

0

@A1 A6 A5

A6 A2 A4

A5 A4 A3

1

A

AIf is symmetric, then we can write

Aij A↵

i, j = 1� 3 ↵ = 1� 6

85PageV. Tournat - Introductory lecture on nonlinear acoustics 856. Analysis of the propagation

Case of the isotropic solids

➪ 3 third order elastic constants

C111 C456C166

➪ 2 second order elastic constants

C11 C44

C11 = C22 = C33

C44 = C55 = C66

C12 = C23 = C13 = C11 � 2C44


The elastic strain energy expansion can be written in several manners using other expressions of the elastic constants and

invariants of the strain tensor

The material behavior : elastic energy as a function of strain


Principal invariants Landau invariants

6. Analysis of the propagation

Relations between the third order elastic constants


Landau & Lifshitz (1986)

Toupin & Bernstein

(1961)

Murnaghan (1951)

Bland (1969)

Eringen & Suhubi (1974) Standard


http://en.wikipedia.org/wiki/Hooke%27s_law

Propagation operator

cl =

sC11

⇢0

L-wave nonlinear term

Plane waves in isotropic solids~U = (U1, U2, U3) ~a = (a1, 0, 0)Propagation along direction

⇢0@2U1

@t2= C11

@2U1

@a21+ (3C11 + C111)

@U1

@a1

@2U1

@a21+ (C11 + C166)

✓@U2

@a1

@2U2

@a21+

@U3

@a1

@2U3

@a21

◆

Nonlinear generation of longitudinal plane waves (component )U1

S-wave nonlinear term

Asynchronous (non cumulative)

! 2! a1

A(2!)


Longitudinal plane waves

@2U1

@t2= c2l

@2U1

@a21

1 +

✓3 +

C111

C11

◆@U1

@a1

�

@2U1

@a21� 1

c2l

@2U1

@t2= �

✓3 +

C111

C11

◆@U1

@a1

@2U1

@a21

@2U1

@a21� 1

c2l

@2U1

@t2= �

✓3

2+

C111

2C11

◆@2U2

1

@a21

�l = �✓3

2+

C111

⇢0c2l

◆Parameter of quadratic nonlinearity for longitudinal waves

U2 = 0 U3 = 0Considering only the longitudinal component


Longitudinal plane waves

@2U1

@a21� 1

c2l

@2U1

@t2= �

✓3

2+

C111

2C11

◆@2U2

1

@a21

Equivalent to the Westervelt equation for fluids


➪ Same methodology than in fluids, Burgers’ equation and its solutions, KZK equation, acoustic diffusivity, wave distortion, shock…



�l

Medium

Duraluminium 5.5

Silica -3.75

Titanium 2.5

BK7 glass -1

�l


Shear plane waves in isotropic solids

⇢0@2U2

@t2= C66

@2U2

@a21+ (C11 + C166)

✓@U1

@a1

@2U2

@a21+

@U2

@a1

@2U1

@a21

◆

➪ Without a longitudinal component ( ),no nonlinear quadratic terms

U1 = 0

ct =

sC66

⇢0➪ The first nonlinear terms for pure shear waves are cubic


Shear plane waves in soft solids


S. Catheline et al., Observation of shock transverse waves in elastic media, Phys. Rev. Lett. 91, 164301 (2003)

Modified Burgers equation

Cubic nonlinear term


Digression on Acousto-elasticity➪ Static stress dependence of the wave speed

through the 3rd order elastic constants

F0

cl,s(F0)

F0


Static force (or quasi-static)

Ultrasonic probing

➪ A method to measure the third order elastic constants

Pulse-echo method

Resonance method

Coda wave interferometry


Digression on Acousto-elasticity


Isotropic solid, propagation along axis 1

"

�i Principal strain components

vij = v0ij(1 + �"ij · ")

Wave velocity when a static strain is applied

�1 = " �2 = �3 = �⌫"Uni-axial loading along axis 1:

L wave

S wave

S wave


Summary

➪ For simplified cases, the tools and equations (Burgers, KZK...) of fluids can be applied

➪ For nonlinear acoustics in solids the Lagrangian description is usual

U = x� a

➪ A propagation equation similar to the Westervelt equation is obtained

⇢0@2Ui

@t2=

@2Uk

@aj@al

✓Cijkl +Mijklmn

@Um

@an

◆

➪ For symmetry reason, no quadratic nonlinearity for pure shear waves

➪ Pure shear waves can interact to generate longitudinal waves but not efficiently (non cumulative, asynchronous)

a1A(2!)


Nonlinear surface waves


A. M. Lomonosov, P. Hess, and A. P. Mayer,Observation of solitary elastic surface pulses,

Phys. Rev. Lett. 88, 076104-1-4 (2002).

Dispersive SAW due to the presence of the thin layer on top of the substrate

Under some conditions, the wave can obey to a KdV equation


Many other effects, processes

➪ Non-collinear interactions, S/L mixing

➪ Heterogeneous media, multiple scattering


➪ Anisotropic solids (QL and QT waves, more couplings, more constants)

➪ Rayleigh, Lamb, Love, Sholte, Stoneley waves…

➪ Plates, bars (geometrical NL)…, buckling, tensegrity

http://bertoldi.seas.harvard.edu/


➪ CR2 : Diffusion multiple (A. Derode)

➪ CR8 : Turbulence d'onde (C. Touzé)

➪ CR6 : Vibrations non linéaires (B. Cochelin)






Fluids




Solid

s



« Mesoscopic solids », nonclassical nonlinearities

Granular media Rocks, sandstones Cracked solids Media with internal contacting surfaces

1017. Nonclassical nonlinearities PageV. Tournat - Introductory lecture on nonlinear acoustics 101

➪ Excitation of large local strains from moderate stresses in the solid matrix

Soft inclusions Defect resonance

Cracks, solid contactsFluid squeezing

➪ A very high level of nonlinearity and different types of nonlinearities

➪ CR9 : Nonlinear acoustic NDE (I. Solodov)

One origin of mesoscopic material strong nonlinearities

Crack

Considering a small nonlinearity for the springs

The quasi-static behavior of the chain is characterized by

-1 0-2-3-5 -4-6-7

0

-1

-2

1

2

Soft inclusion density (Log)

Chain parameter value (Log)


Linear effects

NL effects/

methods

7. Nonclassical nonlinearities

Examples of soft features and non classical behaviors


Frictional contacts and interfaces, short range adhesion, fluid squeezing

➪ Hysteresis➪ Nonlinear dissipation➪ Nonlinear softening

Tapping, clapping contacts and contacting surfaces, local singular stress-strain behavior

➪ CAN➪ Super-harmonics, chaos➪ Sinc modulation spectrum

Delaminations, cracks, imperfectly

contacting surfaces…


Imperfectly contacting surfaces


O. Buck, W.L. Morris, J.M. Richardson, Appl. Phys. Lett. 33, pp. 371 (1978).

➪ 2nd harmonic generation at a solid interface


D.J. Holcomb, Memory, Relaxation and microfracturing in dilatant rock, J. Geophys. Res. 86, 6235-6248 (1981)

➪ Hysteresis in the time of arrival of waves for increasing and decreasing static load

+ conditionning and slow dynamics...

R. Guyer, P. Johnson, Phys. Today 52 (1999)

Hysteresis in an acousto-elastic test

http://www.ees.lanl.gov/ees11/geophysics/nonlinear/nonlinear.shtml

105PageV. Tournat - Introductory lecture on nonlinear acoustics 1057. Nonclassical nonlinearities

Nonlinear resonances

Peculiar processes specific to mesoscopic solids or

damaged solids

➪ Softening

➪ Dissipation ➚

−60

−50

−40

−30

−20

−10

0

10

14000 15000 16000 17000 18000 19000 20000Frequency (Hz)

Am

plitu

de (d

B, a

rb. r

ef.)

Increasing excitation amplitiude


�f

f0=

f(⇥A)� f0f0

= ��f⇥A1

Q(⇥A)� 1

Q0= �

✓1

Q(⇥A)

◆= �Q⇥A

r = ⇥�(1/Q)

|�f/f | = ⇥�Q

�f

Read parameter

Relative frequency shift Shift in inverse quality factor


Odd-symmetry nonlinearity

➪ Preferential generation of odd harmonics (even for L-waves)

➪ Quadratic dynamics in amplitude

G.D. Meegan, Jr., P.A. Johnson, R.A. Guyer, and K. R. McCall, J. Acoust. Soc. Am. 94, 3387 (1993)


Hysteretic loop

�/�A

" = "A cos(✓)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

h > 0

�H = �hE

⇥A⇥+

1

2(⇥2 � ⇥2A)sign(⇥)

�

�H

hE⇥2A

Strain excitation signal

Contribution of hysteresis to stress

Quadratic hysteresis


�W =

I⇥H(�)d⇤(�) > 0

➪ Lost energy in one cycle = surface of the loop

➪ Parabolic curves➪ Odd symmetry

�E = �hE [�A + � sign(�)]

➪ Instantaneous elastic modulus


Phenomenological model of hysteresis: Preisach-Mayergoyz model

F. Preisach, Z. Phys. 57, 3803 (1935)M.A. Krasnosel’skii, A.V. Pokrovskii, “Systems with hysteresis”, Nauka/Springer, Moscow/Berlin, 1983/1989I.D. Mayergoyz, J. Appl. Phys. 57, 3803 (1985), & Phys. Rev. Lett. 56, 1518 (1986)

"1 "2

�2

�1

�

"

Hysteron

Individual hysteretic elements �� = �1 � �2 = cste > 0

➪ Instantaneous transitions

➪ Non dispersive model

➪ Open state / Close state

K.R. McCall, R.A. Guyer, J. Geophys. Res. 99, 23987 (1994)

R.A. Guyer, K.R. McCall, G.N. Boitnott, Phys. Rev. Lett. 74, 3491 (1995)


Phenomenological model of “Preisach-Mayergoyz”"1

"2

�(�1, �2)

�(�1, �2) Distribution of hysterons

�T =Z �2

�1d⇥1

Z +1

�1

�(⇥1, ⇥2, ⇥)�(⇥1, ⇥2)d⇥2

Contribution to the total stress


−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

"

�400 hysterons40 hysterons10 hysterons

Quadratic hysteresis�(�1, �2) = cste

1D nonlinear propagation equation

⇤ = E{⌅+ �⌅2 + ⇥⌅3 � h[⌅A⌅+1

2(⌅2 � ⌅2A)sign(⌅)]}

Nonlinear stress-strain relation in 1D

�0⇤2u

⇤t2=

⇤⇥

⇤xMotion equation / Euler � =

⇥u

⇥xStrain

QNL / ⇥�NL

⇥x➪ Nonlinear source term

Propagation equation⇤2u

⇤x2� 1

c20

⇤2u

⇤t2= � 1

�0c20

⇤⇥NL

⇤x


Hertz nonlinearity� = C"3/2

�2 =1

4�0� 103


�� =32E�⇥1/2

0 �⇥(1 +1

4⇥0�⇥ + . . .)

�0 + �� = C(⇥0 + �⇥)3/2H(⇥0 + �⇥)

uT

= {1 + (1� Tmax

)2/3 � 2

"1� T

max

+ Tsign(T )

2

#2/3

} sign(T )

Hertz-Mindlin nonlinearity in shear motion

➪ At the first order, a quadratic hysteretic nonlinearity

➪ The weaker the contact, the higher its nonlinearity

Contact between two spheres


Noncascade processes in a 3D unconsolidated granular medium

Difference frequency

Non cascade

Pump waves

10 50[kHz]

−70

−60

−50

403020

−100

−90

−40

−80

0

Mag

nitu

de [d

B]

f

σt0

T/2

t

−T/2


Granular chains, sonic vacuum

unun�1 un+1

n+ 1nn� 1

➪ CR3 : Milieux granulaires (G. Theocharis)

➪ Rich (strongly) nonlinear wave phenomena


Compressed granular chains

md2un(t)

dt2= ↵(un+1 � 2un + un�1)�

�

2(un+1 � 2un + un�1)(un+1 � un�1)]

2R

2R-δ0

F0

(a)

(b)

T (f)

f

➪ A nonlinear phononic crystal

!⇤


Second harmonic generation in a granular chain

20 40 60 800.97

0.98

0.99

1

1.01

1st

ha

rmo

nic

0 20 40 60 80 1000

0.05

0.1

0.15

n (bead position)

2n

d h

arm

on

ic

0

0.5

1

1.5

dc

mo

de

20 40 60 80 2 4 6 80

0.1

0.2

0.3

0.4

dc mode

2 4 6 8

0.998

0.999

1

0 2 4 6 8 100

0.01

0.02

0.03

0.04

(bead position)

No

rma

lize

d a

mp

litu

de

Fundamental

Harmonic 2

2! < !⇤ 2! > !⇤

V.J. Sanchez-Morcillo et al. Second harmonic generation for dispersive elastic waves in a discrete granular chain, Phys. Rev. E 88, 043203 (2013).

5. Constitutive equations PageV. Tournat - Introductory lecture on nonlinear acoustics 116PageV. Tournat - Introductory lecture on nonlinear acoustics 1167. Nonclassical nonlinearities

Pure rotational waves in a granular chain

Electro-dynamic shaker

Accelerometers

Fixed bead

A B

19 mm

n=0

n=15

J. Cabaret, P. Béquin, G. Theocharis, V. Andreev, V.E. Gusev, and V. TournatHysteretic Nonlinear Rotational Waves in Granular Chain, Submitted (this week…)

➪ A pure hysteretic nonlinearity


The interaction force: torsional momentGa2

µF0�� =

✓1� 3

2

(M⇤z �Mz)

2µF0a

◆1/2

� 1

2

"1 +

✓1� 3

2

M⇤z

µF0a

◆1/2#� 3

16

Mz

µF0a

Ga2

µF0�+ = �

✓1� 3

2

(M⇤z +Mz)

2µF0a

◆1/2

+1

2

"1 +

✓1� 3

2

M⇤z

µF0a

◆1/2#� 3

16

Mz

µF0a

Increasing moment

Decreasing moment

H. Deresiewicz, Contact of elastic spheres under an oscillating torsional couple, J. of Appl. Mech., 21 :52–56, (1954).

Mn = M linn +Mh

n = �Kt

⇢� + h

�⇤� +

1

2

��2 � �⇤2�sign(�)

��

Kt = d(1� ⌫)F0

At the leading order of nonlinearity, for an oscillating moment and inverting the relation:

h = Ea2/(1 + ⌫)µF0




Linear propagation of rotational waves

50 100 150 200 250 300 350-60

-40

-20

0

20

ModelExperiment

(Hz)fA 15

(dB,

nor

m.)

(Hz)

ModelExperiment

Electro-dynamicshaker

Accelerometer

Fixedbead

19 mm

n=0

n=15

00

100

200

300

(a)

(b)

(c)


Distortion of the pulses in a 70 bead-long chain

0 0.02 0.04 0.06 0.08 0.1

−1

−0.5

0

0.5

1

Time [s]

Acce

lera

tion

[nor

m.]

0.02 0.04 0.06 0.08 0.1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time [s]Ac

celer

ation

[nor

m.]

Increasing excitation level

Signal at bead 1 Signal at bead 18

Signal at ~100Hz

➪ NL velocity decrease

➪ NL attenuation and distortion121PageV. Tournat - Introductory lecture on nonlinear acoustics 1217. Nonclassical nonlinearities

Modeling of the distortion

0 2 4 6 8-2

-1

0

1

2Experimental signalApproximated signal

0 1 2 3 4-1.5

-1

-0.5

0

0.5

1

Normalized timeNo

rmali

zed

veloc

ity

@✓v@⇠

� 1

2

@Mh

@✓v

@✓v@⌧

= 0

Evolution equation for angular velocity:

Moment-Angle relationship calculated with the Preisach-

Mayergoyz formalism

Increasing


A good agreement model-experiments

➪ Observation of pure rotation wave propagation and dispersion

➪ Observation and modeling of the nonlinear distortion of a pulse by a pure quadratic hysteresis

(a) (b)

(c) (d)

0 1 2 3 40 1 2 3 4

-1.5-1

-0.50

0.51

0 0.5 1 1.5 2 2.5

0.60.8

11.21.41.61.8

2

0 0.5 1 1.5 2 2.5

-1.5

-1

-0.5

0

0.5

1

-1.5-1

-0.50

0.51

Norm

alize

d am

plitu

de

Norm

alize

d tim

e


Example of a NDT application

Nonlinear modulation method

➪ Global inspection of the sample, crack detection, monitoring

Coda wave interferometry (CWI) in a multiple scattering medium

Concrete, complex shape systems...

! + � � ! ± �

!

A

! ± ⌦

⌦Selectively sensitive to defects


Analysis of multiply scattered signals

CC(t1,t2)(h0,h1)

(⌧i) =

R t2t1

h0[t(1 + ⌧i)] · h1[t]dtqR t2t1

h20[t(1 + ⌧i)]dt ·

R t2t1

h21[t]dt

Stretching cross-correlation coefficient Stretching parameter

↵ = (v1 � v0)/v0 = �v/v0

CC

(t1,t2)(h0,h1)

(↵) = max[CC

(t1,t2)(h0,h1)

(⌧i)] Kd = 1� CC(↵)

Relative change in velocity Shape modification

2 parameters

⌧i

Coda signal


The method of coda nonlinear modulation

➪ We compare the coda signals received when the pump wave is excited and in the absence of pump wave

> 15 excited resonant modes


Coda nonlinear modulation


Y. Zhang, V. Tournat, O. Abraham, O. Durand, S. Letourneur, A. Le Duff, B. Lascoup, Nonlinear mixing of ultrasonic coda waves with lower frequency-swept pump waves for a global detection of defects in multiple scattering media, J. Appl. Phys. 113, 064905 (2013).

Coda analysis results

Step number

Step number

Step number

IntactDamaged

IntactDamaged

(a)

(b)

(c)

0.002-0.002

-0.014-0.010-0.006(%

)

Kd (%

)0

4

8

12

0 2 4 121086 1614

0 2 4 121086 1614

0 2 4 121086 1614

0

20

40

60(dB)

↵ = (v1 � v0)/v0 = �v/v0

Kd = 1� CC(↵)

Relative velocity change

Shape (attenuation) modification


Numerous possible effects for NDT

NL effect Principle Involved nonlinearities Advantages Drawbacks

Harmonic generation Simple Residual NL

Attenuating media


Robust Weakly sensitive to

residual NL

Attenuating media Insensitive to qudratic

NL

Self-demodulation

Simple High directivity

Attenuating media

Residual NL Non-odd NL

NL modulation Dynamic AE

Weakly sensitive to residual NL

Very different frequencies

Modulation transfer NL absorption Complex to implement

Sub and super-harmonics

CAN Imaging

Require strong amp. Measurement chain

NL

NL CWIComplexe geometries

and media SHM

Requires weak absorption

Pulse inversion Simple Temporal domain Particular cases

s

s

s

s

+ relaxation, slow dynamics, conditioning

! + � � ! ± �

!2 � !1 ! ⌦

! ! !

! + ! ! 2!

!1 ± �+ !2 � !2 ± �

� ! �/2, 3�

To conclude, one thing to remember

Non linéaire

Non-linéarité




[email protected]

Oléron 02/06/2014


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