Remarkable properties of Lamb modes in plates - ACMAC

Women in Applied Mathematics Heraklion, May 2-5, 2011

Remarkable properties of Lamb modes in plates Claire Prada, Daniel Royer, Dominique Clorennec, Franck Philippe Maximin Ces, Todd Murray and Oluwaseyi Balogun,

Institut Langevin, Ondes et Image – ESPCI – CNRS, Paris Department of Electrical and Computer Engineering, Boston University

Local resonance of a plate : impact echo method

d

‘Expected’ Resonance at First longitudinal thickness mode

VL

fL=

2d

but measured resonance at

f
> thickness resonance k = 0 or λ = ∞ group velocity dk

Origin of the local resonance Laser impact (∅ ∅ ≅ 2d) Thermoelastic expansion

Bulk waves

Frequency × thickness (MHz.mm)

Lamb waves

d

S3 dω

A2 S2

S1

=0 Zero Group Velocity : dk Energy is trapped under the source ZGV resonance of S1 mode

S0

A1 A0

VL = 6340 m/s VT = 3100 m/s

Tolstoy et Usdin (JASA vol. 29, 1957): « this point must be associated with a

Thickness / wavelength

kd/2π

sharp CW resonance and ringing effects »

Group Velocity (mm/µs)

Lamb modes and Zero Group Velocities

S0

4

3

S1

A0

4 thickness resonances + S1 minimum frequency resonance

2

1

dω dk

A1

S2

A2

0

1.5VT

0.5VT -1

S2b 1

1.5

2

2.5

3

3.5

4

4.5

5

F.h (MHz.mm)

Where group velocity vanishes a resonance occurs

Pulse laser Experiment Duralumin plate Thickness (d) : 0.49 mm

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

Nd:YAG laser (1064 nm) Pulse duration : 20ns Energie : 4mJ

Pulse laser source couples very well with ZGV resonance Source and detection superimposed

Duralumin plate d = 0.49 mm

Normal surface displacement

Frequency spectrum

Reflected A0 mode

High frequency

Time (µs)

A0 Amplitude (A.U)

Displacement (nm)

Low frequency A0 mode

S1

Frequency (MHz)

S1 mode ZGV resonance Source and detection superimposed


Displacement (nm)

Acquisition time : 4 ms (excitation 10ns)

∆f = 430 Hz Quality factor: Q = 13 400

Time (µs) Frequency (MHz) (MHz) Frequency

Lamb modes measurement Experiment on the 0.5 mm Duralumin plate

Bscan u(r,t)

Moving detection point Detection - step 10µm

Time (µs)

- Ø detection = 30µm

Source: Ø beam = 1mm

Distance (mm)

dB

Lamb modes Measurement Experiment on the 0.5 mm Duralumin plate

dB

Bscan after HP filter u(r,t)

Moving detection point 0

Detection - step 10µm

0 -10

5


TimeTime(µs) (µs)

-20 -30

10

-40 15

-50 -60

20 -70 25


0

2

4 6 Distance (mm)

Distance (mm)

8

10

-80

At the resonances frequency two counter-propagating modes interfere

dB

Frequency (MHz)

Temporal Fourier transform

Distance from the source (mm)

At the resonances frequency two counter-propagating modes interfere

Frequency (MHz)

Temporal Fourier transform

Spatial Fourier transform

Distance (mm) Frequency (MHz)

600

S1

Amplitude (A.U.)

500

S2b

400

300

S2

S2b

S1

S0

200

k/2π π (mm -1)

A0 100

0 -2

S1 mode -1

0

1

2

3

-1

Spatial frequency k/2π (mm )

4

S2b mode

u(x,t) = a1e j(k1x + ω t) + a2be j(k2bx + ω t) with k2b = – k1

-1

6 4 3 2 1 0

-2

Imag(k)*Thickness

From Nicolas Terrien, ONERA

4

3

8

2

7

1

6

0

5

-1

4

-2

Real(k)*Thickness

3

-3

2

-4

1

-5

0

0

-3

0

-4

1

1

-5

2

2

-6

3

3

S2b

4

4

Frequency Thickness MHz.mm

S1

S2

5

5

6

6

5

Complex solutions k of the Rayleigh Lamb equation

Out-of-plane displacement

Position in the plate

ZGV mode is a standing mode d/2

-d/2

Simulations with Spicer model (APL 1990)

In-plane displacement

Position in the plate

At the resonance : ‘Combination of’ a shear thickness mode and a stretch thickness mode

d/2

-d/2 Distance from the laser source (µm)

Sharp resonance Source and detection superimposed


Displacement (nm)

Acquisition time : 4 ms (excitation 10ns)

∆f = 430 Hz Quality factor: Q = 13 400

Time (µs) Frequency (MHz) Frequency (MHz)

Application : Detection of an adhesive disbond Lasers

Air bubble

Duralumin glue glass

C-scan of ZGV resonance Amplitude

Distance (mm)

dB

Distance (mm)

Application : Thickness profile measurement

Thickness (mm)

Profile of the 0.49 mm Duralumin plate

∆d /d = – ∆f /f

Distance (mm) Scanned length: 60 mm

Application : corrosion detection

Plate corroded with orthophosphoric acid solution

1 µm

1.5 µm

10 min

20 min

30 min

Thickness variation (µm)

Sensitivity : 0,1 µm (0,02 %) Resolution : 1 mm (source)

0.5 µm

0.5 µm

1 µm

1.8 µm

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

Outline 1. What is this resonance ? Link with the existence of a backward wave Experimental evidence using optical generation and detection Some applications

2. How does this resonance decay ? Attenuation measurement

3. Are there other resonances of this type ? Local Poisson’s ratio measurement

4. What happens for an anisotropic plate ? Example of silicone wafer

5. Can we play with the backward wave ?….

How does this ZGV resonance decay ? ωd/2π π

Dispersion curves

3

Parabolic approximation around ZGV point:

ω(k ) ≅ ω0 + D(k − k 0 )

2.8

2

S2

2.6 Normal displacement:

1 u ( r,t ) = 2π

+∞

∫

Cth (k )Q (ω ) B (k ) J 0 (kr )e iωt kdk

2.4 2.2

0

Thermoelastic conversion coefficient

(k0,ω0)

Laser source

2

0

0.2

A(k 0 ) 4πDt

J 0 (k 0 r )ei (ω 0t +π / 4)

0.4

kd/2π π

Stationary phase method

u (r,t ) =

S1

with

A(k0) = Cth(k0)Q(ω0)B(k0)k0

Amplitude decreases as t -1/2

The temporal decay provides local attenuation Short time

Long time 0.4

Amplitude (A.U)

0.3

Amplitude at ZGV frequency 0.2

t

0.1

0 0

e

-1/2

-t/τ 20

40

60

80

Time (µs)

u (t ) ∝ t −1 / 2 e − t / τ ⇒ attenuation α

−

=

τ

Prada, Clorennec and Royer, Wave Motion (2008)

τ

α

f (MHz) Copper 4.6

(µ µs) 60

(dB/m) 18

Steel

6.1

400

2.1

Dural

5.8

800

0.9

100

A question for you : The resonnance decay was calculated for lossless medium Symmetrical Lamb modes for a plate with Poisson’s ratio = 0.29 (steel)

lossless medium

lossy medium (attenuation 0.1 np/wl)

How far is the parabolique approximation valid?

Figures from Simonetti and Lowe, JASA 2005






Higher order ZGV modes Frequency × thickness (MHz.mm)

Displacement measured on a Fused silica plate (d = 1,1 mm)

Amplitude (A.U)

25 20 15 10 5 0

2

4

Frequency (MHz)

f2 / f1

6

A2 S1 S1 f2 d = 5,44 MHz.mm A2 f1 d = 2,85 MHz.mm


kd/2π π

2

1.9

Fused silica:

1.8 1.7

ν = 0.172

Absolute and local measurement of Poisson’s ratio

1.6 1.5 0

f2 = 1.905 f1

0.1

0.2

0.3

Poisson's ratio ν

0.4

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

ZGV modes are associated to modes repulsion

VT =

Positions of thickness modes at k = 0 are decisive

1 - 2ν VL 2(1 -ν )

Thickness modes Coincidence of cut-off frequencies for two modes of the same family

Thickness stretch

fcd = VL

VL /2

thickness / transverse wavelength

fcd/VT

5

S10

A6

A9 S8

4

S5

A7 A4

S6

3

Thickness shear

fcd =

A5 S3

2

S4

A2

1

0 0

A3 S2

S1

0.1

0.2

0.3

Poisson's ratio

VT

A1 0.4

0.5

VT

Coincidence of thickness modes at k = 0: • same symetry • different parity

Modes coupling for k ≠ 0

ZGV Modes

Frequency thickness / transverse velocity (fd/V ) T

Thickness modes and ZGV resonances S10

5

A9

A6

S8 S5

4

S8

A7A4

S5

A7

S6S3

3

A4

S6

A5A2

S3

S4

2

A3A2

A2

A3

S2 S1

1 S1

0

A5

S2 A1

0

0.1

0.2 0.3 0.4 Poisson's ratio ν

0.5

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

Local vibration spectrum fd/VT Duralumin ν = 0.338

Fused silica ν = 0.172

S5 S10

S5 S8

S3 S6

A2 A3 S1 S2

S1 S2

Amplitude (dB)

Poisson ratio

Amplitude (dB)






What happens for Anisotropic plates ? cut: [0 0 1] thickness: 0.525mm diameter: 5 ′′

[1 0 0] [1 1 0]

ωd/2π π Frequency × thickness (MHz.mm)

Silicon wafer

[1 0 0] [1 1 0]


On the backward Lamb waves near thickness resonances in anisotropic plates A.L. Shuvalov, O. Poncelet, IJSS 45 (2008)

kd/2π π

Excitation with a point source

Normal displacement spectrum

S1

A0

A2 Frequency (MHz)

Amplitude (A.U)

Amplitude (A.U)

S1 mode

ZGV Cut-off

Frequency (MHz)

Line source

Frequency (MHz)

Displacement spectrum

Laser source

Thickness mode

S1 ZGV mode

S1 ZGV mode

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

Frequency (MHz)

Detection

Angle (degrees)

Dispersion curves (mode S1/S2b) [1 1 0]

0°

[1 0 0]

45°

Phase velocity (km/s)

11.25° 45°

Frequency (MHz)

Frequency (MHz)






Lamb modes measurement Experiment on the 0.5 mm Duralumin plate Moving detection point

Bscan : normal displacement u(r,t)

Detection - step 10µm

Time (µs)



Distance (mm)

dB

Temporal + Spatial Fourier Transforms of u(r,t) provide dispersion curves 40

Absolute Phase Velocity (mm/µs)

35

Negative phase velocities

S2

30

S2b

25 20 15

A1

10

S1

S0 5 A0 1

1.5

2 2.5 3 3.5 4 Frequency x Thickness (MHz.mm)

4.5

5

Hum, there should be something special to do with this backward mode ….

Negative refraction and focusing θi θ r

Medium 1

Medium 2

−θt

sin (θi ) sin (θ r ) sin (θt ) = = v1 v1 v2 If v2 is negative Wave refracts on opposite side of the normal

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

“Veselago Lens” Negative Velocity medium

Can we achieve such a planar lens with Lamb waves?

Negative refraction at a thickness change

thin First experimental evidence with Todd Murray using a continuous modulated laser source

thick

Bramhavar & al. Phys Rev B 2011 Pb : a laser source generates several modes

An array of transducers is used to achieve selective generation of S2 mode Array of 128 transducers

Detection laser

Duralumin plate Fully Programmable parallel process multi-channel electronic device

Experiment by Franck Philippe

Single S2 mode generation Acquisition of the dispersion curve

u(ω,k) Signal transmitted to the array using the programmable electronic device

Mode selection +binarization

u(ω,k)

Experiment by Franck Philippe

2D Inverse Fourier Transform

u(t,x)

Interferometer

42

Temporal frequency analysis of the measured displacement

Focal spot ~λ

Coïncidence frequency

Conclusion • Zero Group Velocity is not a rare phenomenon. It appears in a range of Poisson’s ratio over the value for which the cut-off frequencies of modes belonging to the same family coincide. • Minimum frequency results from the coupling of a pair of modes having different parities, such as S2m+1 and S2n or A2n and A2m+1 • Laser based ultrasonic techniques are very efficient for investigating specific properties of ZGV Lamb modes: - resonance and ringing effects - interference between backward and forward waves. - backward wave propagation • Experiments show that the local resonance spectrum of an unloaded elastic plate is dominated by the ZGV Lamb modes. Accurate local material characterization without any mechanical contact

Experimental evidence of S1-ZGV resonance • Holland and Chimenti (Appl. Phys. Lett. vol. 83, 2003) observed with air coupled transducers the transparency of a plate due to the S1 mode ZGV resonance

• Gibson and Popovics (J. Eng. Mech. vol. 131, 2005) Explained the shift observed on the resonance frequency of a concrete plate

• Prada, Balogun and Murray (Appl. Phys. Lett. vol. 87, 2005) CW Laser generation and detection of ZGV resonance on 50 µm thick tungsten plates

• Clorennec, Prada, Royer and Murray (Appl. Phys. Lett. vol. 89, 2006) Pulse Laser generation and detection of the ZGV resonance

Other references I. Tolstoy and E. Usdin, “Wave propagation in elastic plates: low and high mode dispersion,” J. Acoust. Soc. Am. 29(1) 37 (1958). A.H. Meitzler, “Backward wave transmission of stress pulses in elastic cylinders and plates,” J. Acoust. Soc. Am. 38, 835 (1965). Shuvalov, Poncelet, 'On the backward Lamb waves near thickness resonances in anisotropic plates ‘,Int. J.Solids Structures 45 (2008) Prada, Clorennec and Royer, ''Local vibration of an elastic plate and zero-group velocity Lamb modes'', J. Acoust. Soc. Am. 124, (1) (2008) Prada, Clorennec, Murray, Royer, ''Influence of the anisotropy on zero-group velocity Lamb modes'', J. Acoust. Soc. Am. 126 (2), (2009). Bramhavar, Prada, ..…., Murray, "Negative Refraction and Focusing of Lamb Waves at an Interface", Physical Review B. 83, 014106 (2011).