Propriétés remarquables de certains modes de Lamb Applications€¦ · Propriétés remarquables de certains modes de Lamb Applications Claire Prada avec Daniel Royer, Dominique

Propriétés remarquables de certains modes de Lamb

Applications

Claire Prada avec Daniel Royer, Dominique Clorennec,

Franck Philippe, Maximin Ces et Jérôme Laurent

Institut Langevin, Ondes et Image – ESPCI – CNRS, Paris

Journée scientifique de la Fed3G Mardi 11 Juin 2013

Local resonance of a plate : impact echo method

Impact Echo method for testing concrete structures in civil engineering

(NIST, Cornell University, Sansalone 1986).

• In 1998 an empirical correction factor b=0.96 was introduced in ASTM* C 1383

standard for measuring the thickness of concrete plates.

Apparent longitudinal velocity bVL and resonance at f = b V L

2d

* American Society for Testing and Materials

‘Expected’ Resonance at V L

2d fL=

d

V L

2d f < but measured resonance at

First longitudinal thickness mode

Pulsed Laser excitation of a plate

Duralumin plate

Thickness d = 500µm

Longitudinal velocity VL = 6.34 mm/µs,

Frequency content of the normal surface displacement

Below the thickness

resonance V L

2d

f2d 6 < VL

Laser source Interferometer

Quelques questions

1. Quelle est cette résonance ?

2. Que faire de cette résonance ?

3. Comment cette résonance décroit-elle ?

4. Existe-t-il d’autres résonances de ce type ?

5. Qu’en est-il pour les plaques anisotropes ?

Nd:YAG laser (1064 nm)

Pulse duration : 20ns

Energie : 4mJ

Heterodyne interferometer BW : 20kHz - 40 MHz (532 nm)

Duralumin plate Thickness (d) : 0.49 mm

Pulse laser Experiment

Pulse laser source couples very well into a narrow resonance

Normal surface displacement

Low frequency mode

Reflected mode

High frequency

Dis

pla

cem

en

t (n

m)

Time (µs)

Duralumin plate d = 0.49 mm

Source and detection

superimposed

Am

pli

tud

e (

A.U

)

Frequency (MHz)

Frequency spectrum

Acquisition time : 4 ms (excitation 10ns)

Dis

pla

cem

ent

(nm

)

Time (µs)

…… a very narrow resonance indeed!

f = 430 Hz

Quality factor:

Q = 13 400

Frequency (MHz)

Duralumin plate d = 0.49 mm

Source and detection

superimposed

Detection

- step 10µm

- Ø detection = 30µm

Source:

Ø beam = 1mm

Moving detection point

Experiment on the 0.5 mm Duralumin plate

propagation measurement

dB

Tim

e (µ

s)

Distance (mm)

Bscan u(r,t)

Detection

- step 10µm

- Ø detection = 30µm

Source:

Ø beam = 1mm

Scanning detection point

Experiment on the 0.5 mm Duralumin plate

Propagation measurement

-80

-70

-60

-50

-40

-30

-20

-10

0

Distance (mm)

Tim

e (

µs)

0 2 4 6 8 10

0

5

10

15

20

25

dB

Tim

e (µ

s)

Distance (mm)

Bscan after HP filter u(r,t)

Fre

qu

en

cy (

MH

z)

Distance from the source (mm)

dB

At the resonance frequency a standing wave is observed

Temporal Fourier transform u(r,)

Fre

qu

en

cy (

MH

z)

Fre

qu

en

cy (

MH

z)

Distance (mm)

At the resonance frequency a standing wave is observed

Temporal Fourier transform u(r,)

Amplitude Profile at f = 5.89 MHz

- J0(kx)

Symmetrical modes Antisymmetrical modes

S0, S1 … A0, A1 …

h

x2 x2

A local impact generates Lamb waves

Rayleigh Lamb

equation

2

2

22 k

Vp

L

=2

2

22 k

Vq

T

=

,)(

)(14 22

2

4

=

qhtg

phtg

q

pqk

VT

2

,0

Thermoelastic

expansion

bulk waves

Lamb modes

Laser pulse

Lamb waves are dispersive modes

Fre

qu

en

cy

th

ick

ne

ss

(MH

z.m

m)

Dispersion curves (k)

Thickness / wavelength

A3

S0

A1

S1

A0

S2

S3

VL = 6340 m/s

VT = 3100 m/s

0.5VT

VT

VL

1.5VT

At cut-off frequencies

Stretch or shear vibration

uniform over the plate

k = 0 or l =

duralumin plate

resonance thickness0 velocitygroup =dk

d

Origin of the local resonance

Thermoelastic

expansion Bulk waves

Laser impact ( 2d)

Lamb waves

A2

S0

A1

S2

A0

S1

S3

VL = 6340 m/s

VT = 3100 m/s

Fre

qu

en

cy

th

ick

ne

ss

(MH

z.m

m)

kd/2

d


Tolstoy et Usdin (JASA vol. 29, 1957):

« this point must be associated with a

sharp CW resonance and ringing effects »

Zero Group Velocity :

Energy is trapped under the source

ZGV resonance of S1 mode

0=dk

d

Complex solutions k of the Rayleigh Lamb equation

Real(k)*Thickness

0

1

2

3

4

5

6

-5

-4

-3

-2

-1

0

1

2

3

4

5

Imag(k)*Thickness

0

1

2

3

4

5

6

-1 -2 -3 -4 -5 -6 -7 -8

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 F

req

uen

cy T

hic

kn

ess M

Hz.m

m

S1 S2

S2b

S2b and S2 branches are linked through a purely imaginary branch

-2 -1 0 1 2 3 40

100

200

300

400

500

600

Spatial frequency k/2 (mm-1

)

Am

plit

ud

e (

A.U

.)

S1

S2b

S0

A0

Spatial Fourier transform

Fre

qu

en

cy (

MH

z)

Distance (mm)

At the resonance frequency two counter-propagating modes interfere

Temporal Fourier transform

u(x,t) = a1e j(k1x + t) + a2be j(k2b

x + t) with k2b = – k1

S1 mode S2b mode

k/2 (mm ) -1

Fre

qu

en

cy (

MH

z)

S2b S1

S2

Displacements distribution inside the plate

In-plane

displacement

Out-of-plane

displacement

Simulations

with Spicer model

(APL 1990)

Distance from the laser source (µm)

Po

sit

ion

in

th

e p

late

P

osit

ion

i

n t

he

pla

te d/2

-d/2

d/2

-d/2

At the resonance : ‘Combination of’

a shear thickness mode

and a stretch thickness mode

Quelques questions 1. Quelle est cette résonance ? Elle résulte de l’interférence de deux modes de Lamb à vitesse de groupe nulle (ZGV)

et de vitesse de phase opposées, S1 et S2b, à la fréquence minimum du mode S1





6. Peut-on jouer avec l’onde à vitesse de phase négative ?

Application 1 : Detection of an adhesive disbond

Air bubble Lasers

C-scan of ZGV Resonance Amplitude

dB

Distance (mm)

Dis

tan

ce (

mm

)

Duralumin

glue

glass

Line Scan Across Support

Imaging of Sub-Surface Features

in 4mm thick membranes

350mm

scan

Balogun, Murray, Prada, J. App. Phys., 102 1 (2007)

Amplitude of the ZGV resonance

Murray, Balogun, “High sensitivity laser based

acoustic microscopy using a modulated laser source,”

Appl. Phys. Lett. 85 (14), pp. 2974 (2004).

High frequency measurements

Application 1

Application 2 : Thickness measurement, Corrosion detection

0.5 µm 1 µm 1.5 µm

Plate corroded with

orthophosphoric

acid solution

10 min 20 min 30 min

Sensitivity : 0,1 µm (0,02 %)

Resolution : 1 mm (source)

Th

ick

ne

ss

va

ria

tio

n (

µm

)

Distance (mm)

1 mm

1.8 mm

0.5 mm

Clorennec, Prada et Royer, Appl. Phys. Lett. (2006)

d /d = – f /f

Substrate + layer

Substrate

f

k=0 k=kZGV k

fZGV

fc

Application 3 : Thin layer measurements

m

m K

f

f

zgv

zgv =

m

m

f

f

c

c =

similar to quartz resonator

ZGV resonance and mass sensor

Substrate Layer K

Dural Gold 1.16

Dural Silicon carbide 0.3

Copper Silicon carbide - 0.48 !!

• Thickness resonance

• ZGV resonance

Examples of Numerical values

Ces, Clorennec, Royer, Prada, Review of Scientific Instrument 2011

2%m

m

pour

1.503 mm 100 nm

5 cm scan

frequency

Thickness

• reference scan and

temperature control

required

1.503 mm

5 cm scan

Application 3 : Detection of 100nm of gold through 1.5mm thick plate

Distance (mm)

Fre

qu

en

cy (

MH

z)

S1S2

0 5 10 15 20 25 30 35 40 45 50 1.908

1.909

1.91

1.911

Fre

qu

en

cy s

hif

t (k

Hz)

Distance (mm)

0 5 10 15 20 25 30 35 40 45 50 -2

-1

0

1

2

Δf=0.96 kHz

ZGV

ZGV

f

fmK

m (gold on Dural : K=1.16) >> deposited mass




Mesurer sans contact mécanique l’épaisseur, la masse déposée, un décollement …





200)( kkDωkω

How does this ZGV resonance decay ?

Parabolic approximation around ZGV point:

0 0.2 0.4 2

2.2

2.4

2.6

2.8

3

S 2

S 1

(k0,0)

Dispersion curves

Thermoelastic conversion coefficient Laser source

Normal displacement:

kdkekrJkBQkCr,tu tith

=

0

0)()()(2

1)(

)4/(

4)( 0

000

=

tierkJ

Dt

kAr,tu with A(k0) = Cth(k0)Q(0)B(k0)k0

Amplitude decreases as t -1/2

Stationary phase approximation kd/2

d/2

Application 4 : The temporal decay provides local attenuation

Prada, Clorennec and Royer, Wave Motion (2008)

Short time Long time

/2/1)( tettu

attenuation

0

1 1)(V

m =

f (MHz) (ms) (dB/m)

Copper 4.6 60 18

Steel 6.1 400 2.1

Dural 5.8 800 0.9

0 20 40 60 80 100 0

0.1

0.2

0.3

0.4

Time (µs)

0 20 40 60 80 100

e -t/

t -1/2

Amplitude at ZGV frequency

Am

pli

tud

e (

A.U

)



2. Que faire de cette résonance ? Mesure locale sans contact mécanique d’un décollement, de l’épaisseur, de la masse déposée…


Application : mesure de l’atténuation




/2/1)( tettu

Existence of ZGV modes depends on Poisson’s ratio

Relative positions of thickness modes at k = 0 are decisive

f d =

VT

VL /2

Thickness modes

2

2

2

2 1

L T

L T

V V

V V

=

Application 5 : The two first ZGV modes provide Poisson’s ratio

Clorennec, Prada et Royer, J.Appl.Phys. Vol. 101 (2007)

0 2 4 6

5

10

15

20

25

Frequency (MHz)

S1

A2

Displacement measured on a

Fused silica plate (d = 1,1 mm)

A2

S1

f1 d = 2,85 MHz.mm

f2 d = 5,44 MHz.mm

Fre

qu

en

cy

th

ick

ne

ss

(MH

z.m

m)

Thickness / wavelength kd/2

Am

pli

tud

e (

A.U

)

Absolute and local

measurement

of Poisson’s ratio

Fused silica: = 1.905 =0.172f 2

f 1

0 0.1 0.2 0.3 0.41.5

1.6

1.7

1.8

1.9

2

f2 / f1

Poisson's ratio

Thickness modes

Thickness shear Thickness stretch

f d =

VL

VL /2

c

f d =

VT

VT

c

Coincidence of cut-off frequencies

for two modes of the same family

0 0.1 0.2 0.3 0.4 0.5 0

1

2

3

4 th

ickn

ess /

tra

nsvers

e w

avele

ng

th

Poisson's ratio

A 6

S 5

A 4

A 9

S 1 0

S 8

A

7

A

5

S 4

A 1

A 3

S 1

S 2

S 6

S 3

A

2

5 f d/VT c

Fre

qu

en

cy t

hic

kn

es

s / t

ran

sve

rse

ve

loc

ity (

fd/V

)

T

0 0.1 0.2 0.3 0.4 0.5 0

1

2

3

4

5

Poisson's ratio

A 6

S 5

A 4

S 3

A 2

S 1

A 9

S 1 0

S 8

A

7

S 6

A

5

S 4

A

3

S

2

A

1

A 2

A 2

A 3

S

2

S 1

A 5

S 6

S 3

S 5

S 8

A 4

A 7

Thickness modes and ZGV resonances

Coincidence of thickness

modes at k = 0:

• same symetry

• different parity

Modes coupling

for k 0

ZGV Modes

Prada, Clorennec et Royer, J. Acoust .Soc. Am. Vol.124 (2008)

Thickness / wavelentgh (kd / 2)

0 0.5 1 3.8

3.9

4

4.1

4.2

4.3

4.4

4.5

= 0.155

0 0.5 1 3.8

3.9

4

4.1

4.2

4.3

4.4

4.5

= 0.179

0 0.5 1 3.8

3.9

4

4.1

4.2

4.3

4.4

4.5 = 0.200

0 0.5 1 3.8

3.9

4

4.1

4.2

4.3

4.4

4.5

f d

/ V

T

= 0.130

S 5

S 8

S 5

S 8

S 5

S 8 S

5

S 8

k 0

Coïncidence Dispersion curves

Mode repulsion : case of S5/S8

Local vibration spectrum provides Poisson’s ratio

Fused silica = 0.172 Duralumin = 0.338

Poisson ratio Amplitude (dB) Amplitude (dB)

S1 S2

A2 A3

S5 S8

S1 S2

S3 S6

S5 S10

fd/VT

Local vibration spectrum


et de vitesses de phases opposées, S1 et S2b, à la fréquence minimum du mode S1

2. Que faire de cette résonance ? Mesure locale sans contact mécanique d’un décollement, de l’épaisseur, de la masse déposée…



4. Existe-t-il d’autres résonances de ce type ? Oui, plein! Quand (n+0.5)VL et mVT ou bien nVL et (m+0.5)VT sont proches (n,m entiers)

>> application : mesure sans contact du coefficient de Poisson



/2/1)( tettu

Silicon wafer

cut: [0 0 1]

thickness: 0.525mm

diameter: 5

On the backward Lamb waves near thickness resonances in anisotropic plates

A.L. Shuvalov, O. Poncelet, IJSS 45 (2008)

Anisotropic plates

[1 0 0]

[1 1 0]


d/2

Fre

qu

en

cy

th

ick

ne

ss

(MH

z.m

m)

kd/2

[1 0 0]

[1 1 0]

Excitation with a point source

S1

A0 Am

pli

tud

e (

A.U

)

Frequency (MHz)

Normal displacement spectrum

Frequency (MHz)

Am

pli

tud

e (

A.U

) ZGV Cut-off

S1 mode

A2

Fre

qu

en

cy (

MH

z)

Angle (degrees)

Laser

source

Displacement

spectrum

S1 ZGV mode

Line source

Fre

qu

en

cy (

MH

z)

S1 ZGV mode

Thickness mode

Prada, Clorennec, Murray and Royer, J. Acoust .Soc. Am. Vol.126 (2009)

Detection

Similar behaviour in rolled stainless steel

Angle (°)

Fre

quen

cy (

MH

z)

Frequency (MHz)

Measurement of anisotropy

Measurement of the elastic constants of a zircalloy cladding tube

© La médiathèque EDF / Patrick Landmann

Heterodyne

interferometer

2b

d Pressurized Water Reactors

(PWR) - Zirconium 98%; étain 1,5 %; fer 0,2%; chrome 0,1%

- External diameter 9,5 mm

- Thickness 0,57 mm

M. Cès, D. Royer, C. Prada, J. Acoust. Soc. Am. 132 (1), (2012)

The ZGV resonance spectrum depends on source orientation

axial circonferential

The medium is anisotropic

With isotropic assumption : VL1 = 4740 m/s

VT1 = 2405 m/s

VL2 = 4830 m/s

VT2 = 2300 m/s

Influence of curvature on ZGV resonance frequency

Lamb modes in a plate of thickness d

Circonferential modes in a tube of

external radius a, internal radius b=a-d

thick tube a = 5d (b/a=0.8)

thin tube a = 21d (b/a= 0.95)

For zircaloy tube (b/a=0.87) material anisotropy dominates

o

circonferential

+ axial

10-2

10-4

ZGV frequency slightly modified by curvature

For zircalloy cladding tube a/d ≈ 8

ΔfZGV/fZGV<10-3 for circonferential modes

ΔfZGV/fZGV induced by curvature

n = 0,34

c11 and c66 deduced from circumferential ZGV resonance spectrum

Propagation in ┴x3 plane :

- longitudinal : VL1 = (c11/r½

- transverse polarised ┴x3 : VT1= (c66/r) ½

Isotropy in plane ┴x3

ZGV frequencies provide :

c11 = 148 GPa

c66 = 38 GPa

Circ. measured dispersion curves

Assumption : transverse isotropic medium

c33 and c44 deduced from axial ZGV resonance spectrum

Axial modes depend on

- longitudinal velocity along x3 : VL2 = (c33/r½

- longitudinal velocity along x1 : VL1 = (c11/r½

- transverse velocity with // polarisation :

VT2 = (c44/r) ½

Assuming the velocities are correctly

determined by ZGV method

c33 = 154 GPa

c44 = 35 GPa

c13 is determined by dispersion curves fitting.

Axial measured dispersion curves



2. Que faire de cette résonance ? Mesure locale sans contact mécanique de l’épaisseur, la masse déposée, un décollement …





5. Qu’en est-il pour les plaques anisotropes ? Les modes ZGV existent, ils peuvent être sélectionnés en utilisant un ligne source.

>> mesure locale sans contact de l’anisotropie.


/2/1)( tettu

-2 -1 0 1 2 3 40

100

200

300

400

500

600

Spatial frequency k/2 (mm-1

)

Am

plit

ud

e (

A.U

.)

S1

S2b

S0

A0

Spatial Fourier transform

Fre

qu

en

cy (

MH

z)

Distance (mm)

The S2b mode has a negative phase velocity

Temporal Fourier transform

u(x,t) = a1e j(k1x + t) + a2be j(k2b

x + t) with k2b = – k1

S1 mode S2b mode

k/2 (mm ) -1

Fre

qu

en

cy (

MH

z)

S2b S1

S2

Negative Refraction and Focusing of Backward Waves V.G. Veselago, Soviet Phys. Uspekhi, 10 (1968)

Medium 1

Medium 2

i r

If v2 is negative

Wave refracts on opposite side of the normal t

“Veselago Lens”

Negative Velocity medium

Can we achieve such a planar

lens with Lamb waves?

Negative refraction and focusing

1 1 2

sin sin sini r t

v v v

= =

Negative refraction at a thickness change

thin

thick First experimental evidence with

Todd Murray using a continuous

modulated laser source Bramhavar & al. Phys Rev B 2011

Pb : a laser source generates several modes

Results after low Pass Filter

Quelques questions et réponses 1. Quelle est cette résonance ? Elle résulte de l’interférence de deux modes de Lamb à vitesse de groupe nulle (ZGV)


2. Que faire de cette résonance ? Mesure locale sans contact mécanique de l’épaisseur, la masse déposée, un décollement …





5. Qu’en est-il pour les plaques anisotropes ? Les modes ZGV existent, ils peuvent être sélectionnés en utilisant un ligne source.

>> mesure locale sans contact de l’anisotropie.

6. Peut-on jouer avec l’onde à vitesse de phase négative ? Oui

D’autres questions ?

/2/1)( tettu

Documents

Propriétés remarquables de certains modes de Lamb Applications€¦ · Propriétés remarquables de certains modes de Lamb Applications Claire Prada avec Daniel Royer, Dominique