Remarkable properties of Lamb modes in plates

Transcription

1 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

2 Local resonance of a plate : impact echo method d Expected Resonance at First longitudinal thickness mode V f L L = 2d but measured resonance at V f < L 2d Impact Echo method for testing concrete structures in civil engineering (NIST, Cornell University, Sansalone 1986). In 1998 an empirical correction factor β=0.96 standard for measuring the thickness of concrete plates. Apparent longitudinal velocity βv L and resonance at f = β V L 2d β=0.96 was introduced in ASTM* C 1383 * American Society for Testing and Materials

3 Pulsed Laser excitation of a plate Laser source Interferometer Duralumin plate Thickness d = 500µm Bulk velocities V L = 6.34 mm/µs, V T = 3.10 mm/µs Frequency content of the normal surface displacement Looks like a thickness resonance V L 2d But f2d 6 < V L

4 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?.

5 A local impact generates Lamb waves Thermoelastic expansion Laser pulse Lamb modes bulk waves Rayleigh Lamb equation 4 ω 2 V T q 2 = 4k 2 = ω V 2 T 2 q 2 k 1 2 p 2 p q tg( ph + α), tg( qh + α) 2 = ω V 2 L k 2 α 0, π 2 ω frequency, k wave number, V L longitudinal velocity, V T shear velocity Horace Lamb x 2 x 2 h Symmetrical modes S0, S1 A0, A1 Antisymmetrical modes

6 Lamb waves are dispersive modes At cut-off frequencies Dispersion curves ω(k) 1.5VT VL VT Frequency thickness (MHz.mm) S 3 A 3 S 1 A 1 S 2 A 0 S 0 duralumin plate V L = 6340 m/s V T = 3100 m/s 0.5VT Stretch or shear vibration uniform over the plate dω k = 0 or λ = group velocity = 0 >> dk Thickness / wavelength thickness resonance

7 Origin of the local resonance Thermoelastic expansion Laser impact ( 2d) Bulk waves Lamb waves d Frequency thickness (MHz.mm) S 3 A 2 S 2 A 1 S 1 S 0 A 0 V L = 6340 m/s V T = 3100 m/s dω Zero Group Velocity : = 0 dk Energy is trapped under the source ZGV resonance of S 1 mode Tolstoy et Usdin (JASA vol. 29, 1957): «this point must be associated with a Thickness / wavelength kd/2π sharp CW resonance and ringing effects»

8 Lamb modes and Zero Group Velocities dω dk Group Velocity (mm/µs) S 0 S 1 A 0 A 1 S 2 A 2 4 thickness resonances + S1 minimum frequency resonance 0.5V T 1.5V T -1 S 2b F.h (MHz.mm) Where group velocity vanishes a resonance occurs

9 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

10 Pulse laser source couples very well with ZGV resonance Source and detection superimposed Duralumin plate d = 0.49 mm Normal surface displacement Frequency spectrum Low frequency A 0 mode A 0 Displacement (nm) Reflected A 0 mode High frequency Amplitude (A.U) S 1 Time (µs) Frequency (MHz)

11 S 1 mode ZGV resonance Source and detection superimposed Duralumin plate d = 0.49 mm Acquisition time : 4 ms (excitation 10ns) Displacement (nm) f = 430 Hz Quality factor: Q = Time (µs) Frequency Frequency (MHz) (MHz)

12 Lamb modes measurement Experiment on the 0.5 mm Duralumin plate Moving detection point Detection - step 10µm - Ø detection = 30µm Bscan u(r,t) db Time (µs) Source: Ø beam = 1mm Distance (mm)

13 Lamb modes Measurement Experiment on the 0.5 mm Duralumin plate Moving detection point Detection - step 10µm - Ø detection = 30µm 0 5 Bscan after HP filter u(r,t) db Time Time (µs) Source: Ø beam = 1mm Distance (mm) Distance (mm) -80

14 At the resonances frequency two counter-propagating modes interfere Temporal Fourier transform db Frequency (MHz) Distance from the source (mm)

15 At the resonances frequency two counter-propagating modes interfere Temporal Fourier transform Frequency (MHz) Amplitude (A.U.) Spatial Fourier transform S1 S2b S0 A0 Distance (mm) Frequency (MHz) S 2 S 2b S 1 S 1 mode S 2b mode k/2π (mm -1) Spatial frequency k/2π (mm -1 ) u(x,t) = a 1 e j(k 1 x + ω t) + a 2b e j(k 2b x + ω t ) with k 2b = k 1

16 Complex solutions k of the Rayleigh Lamb equation S 2 S 1 S 2b Real(k)*Thickness Frequency Thickness MHz.mm Imag(k)*Thickness 4 From Nicolas Terrien, ONERA

17 ZGV mode is a standing mode Out-of-plane displacement Position in the plated/2 -d/2 Simulations with Spicer model (APL 1990) At the resonance : Combination of In-plane displacement Position in the plate d/2 -d/2 Distance from the laser source (µm) a shear thickness mode and a stretch thickness mode

18 Sharp resonance Source and detection superimposed Duralumin plate d = 0.49 mm Acquisition time : 4 ms (excitation 10ns) Displacement (nm) f = 430 Hz Quality factor: Q = Time (µs) Frequency (MHz) Frequency (MHz)

19 Application : Detection of an adhesive disbond Lasers Air bubble Duralumin glue glass C-scan of ZGV resonance Amplitude db Distance (mm) Distance (mm)

20 Application : Thickness profile measurement Profile of the 0.49 mm Duralumin plate Thickness (mm) d/d = f /f Distance (mm) Scanned length: 60 mm

21 Application : corrosion detection Plate corroded with orthophosphoric acid solution 0.5 µm 1 µm 1.5 µm 10 min 20 min 30 min Sensitivity : 0,1 µm (0,02 %) Thickness variation (µm) 0.5 µm 1µm 1.8 µm Resolution : 1 mm (source) Distance (mm) Clorennec, Prada et Royer, Appl. Phys. Lett. (2006)

22 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?.

23 How does this ZGV resonance decay? Parabolic approximation around ZGV point: ωd/2π 3 Dispersion curves 0 ( ) 2 ω( k) ω + D k k S 2 Normal displacement: 2.6 u( r,t) = + 1 C 2π 0 th ( k) Q( ω) B( k) J 0 iωt ( kr) e kdk (k 0,ω 0 ) S 1 Thermoelastic conversion coefficient Laser source Stationary phase method kd/2π ( ) A k0 ( ) ( 0 u( r,t) = J i ω t+ π / 4) 0 k0r e with A(k 0 ) = C th (k 0 )Q(ω 0 )B(k 0 )k 0 4πDt Amplitude decreases as t -1/2

24 The temporal decay provides local attenuation Short time Long time 0.4 Amplitude (A.U) Amplitude at ZGV frequency t -1/2 u ( t ) t 1 / 2 attenuation t / τ α = τ Prada, Clorennec and Royer, Wave Motion (2008) e e -t/τ Time (µs) f τ α (MHz) (µs) (db/m) Copper Steel Dural

25 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

26 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?.

27 Higher order ZGV modes Amplitude (A.U) Displacement measured on a Fused silica plate (d = 1,1 mm) Frequency thickness (MHz.mm) A 2 S 1 S 1 f 2 d = 5,44 MHz.mm A 2 f 1 d = 2,85 MHz.mm f 2 / f 1 2 Frequency (MHz) Thickness / wavelength kd/2π Poisson's ratio ν f 2 Fused silica: = ν = f 1 Absolute and local measurement of Poisson s ratio Clorennec, Prada et Royer, J.Appl.Phys. Vol. 101 (2007)

28 ZGV modes are associated to modes repulsion V T = 1-2ν 2 1- V L ( ν ) Positions of thickness modes at k = 0 are decisive

29 Thickness modes Coincidence of cut-off frequencies for two modes of the same family f d/v T c 5 A 6 S 10 f d = c V L V L /2 Thickness stretch thickness / transverse wavelength S 5 S 3 A 4 A 9 S 8 A 7 S 6 A 5 S 4 A 2 A 3 S 1 S 2 Thickness shear f d = c V T V T A Poisson's ratio

30 Thickness modes and ZGV resonances 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 5 A 6 4 S 5 3 A 4 S 3 2 A 2 1 S 1 A 3 A 2 S 5 S 8 A 4 A 7 S 6 S 3 S 2 S 1 A A Poisson's ratio ν S 10 A 9 S 8 A 7 S 6 A 5 S 4 A 3 S 2 A 1 Prada, Clorennec et Royer, J. Acoust.Soc. Am. Vol.124 (2008)

31 Local vibration spectrum fd/v T Fused silica ν = Duralumin ν = S 5 S 10 S 5 S 8 S 3 S 6 A 2 A 3 S 1 S 2 S 1 S 2 Amplitude (db) Poisson ratio Amplitude (db)

32 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?.

33 What happens for Anisotropic plates? Silicon wafer cut: [0 0 1] thickness: 0.525mm diameter: 5 [1 0 0] [1 1 0] ωd/2π Frequency thickness (MHz.mm) [1 0 0] [1 1 0] Thickness / wavelength kd/2π On the backward Lamb waves near thickness resonances in anisotropic plates A.L. Shuvalov, O. Poncelet, IJSS 45 (2008)

34 Excitation with a point source Normal displacement spectrum S 1 mode S 1 Amplitude (A.U) A 0 Frequency (MHz) A 2 Amplitude (A.U) ZGV Cut-off Frequency (MHz)

35 Line source Laser source Displacement spectrum Frequency (MHz) Thickness mode S 1 ZGV mode Detection S 1 ZGV mode Frequency (MHz) Prada, Clorennec, Murray and Royer, J. Acoust.Soc. Am. Vol.126 (2009) Angle (degrees)

36 Dispersion curves (mode S 1 /S 2b ) [1 1 0] 0 [1 0 0] Phase velocity (km/s) Frequency (MHz) Frequency (MHz)

37 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?.

38 Lamb modes measurement Experiment on the 0.5 mm Duralumin plate Moving detection point Detection - step 10µm - Ø detection = 30µm Bscan : normal displacement u(r,t) db Time (µs) Source: Ø beam = 1mm Distance (mm)

39 Temporal + Spatial Fourier Transforms of u(r,t) provide dispersion curves 40 Phase Velocity (mm/µs) Absolute A 0 S 2b S 2 S 0 A 1 S 1 Negative phase velocities Frequency x Thickness (MHz.mm) Hum, there should be something special to do with this backward mode.

40 Negative refraction and focusing θi θr Medium 1 ( θ ) ( θ ) ( θ ) sin sin sin = = v v v i r t θ t Medium 2 Negative Refraction and Focusing of Backward Waves V.G. Veselago, Soviet Phys. Uspekhi, 10 (1968) Veselago Lens If v 2 is negative Wave refracts on opposite side of the normal Negative Velocity medium Can we achieve such a planar lens with Lamb waves?

41 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 An array of transducers is used to achieve selective generation of S 2 mode Array of 128 transducers Detection laser Duralumin plate Fully Programmable parallel process multi-channel electronic device Experiment by Franck Philippe

42 Single S 2 mode generation Experiment by Franck Philippe Acquisition of the dispersion curve Mode selection +binarization 2D Inverse Fourier Transform u(ω,k) u(ω,k) u(t,x) Signal transmitted to the array using the programmable electronic device Interferometer 42

43 Temporal frequency analysis of the measured displacement Focal spot ~λ Coïncidence frequency

44 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 S 2m+1 and S 2n or A 2n and A 2m+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

45 Experimental evidence of S 1 -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, (2011).