Laser ultrasonics for material characterization and defect...
Transcript of Laser ultrasonics for material characterization and defect...
Laser ultrasonics for material characterization and defect detection
Christ GlorieuxLaboratory of Acoustics
Soft Matter and Biophysics
Department of Physics and Astronomy
KU Leuven, Belgium
Alexander Graham Bell & Charles Sumner, 1880Godfather of photoacoustics
Photothermal and photoacoustic phenomena⇒ optical, thermal and elastic properties
photothermal effect
T
z
sinusoidal SAW
gaussian SAW burst SAW
photoacoustic effect
µ
⇒ information on transport properties: ⇒ thermal diffusivity/diffusion length& acoustic velocity and damping/wavelength
λ
Photoacoustic tomography: optical information
Overview
• Laser ultrasonics in layered samples: elastic/thermal depth information from dispersion
• Optical detection schemes• Michelson interferometer• Phase mask interferometer• Sagnac interferometer• Laser beam deflection • Speckle knife edge detection (SKED)• Laser Doppler vibrometry• Photorefractive interferometer• Modulated optical reflection• Brillouin oscillations
• Laser ultrasonics for material characterization: case studies• Calculation of guided wave dispersion and photothermal and photoacoustic displacements
• Laser ultrasonics for defect detection and application for non-destructive testing
Laser ultrasonics on layered samples:
Elastic/thermal depth information from dispersion
vR1
vR2
vR3
Rayleigh waves: wavelength dependent penetration depth
Laser ultrasonics on layered samples: elastic information from guided wave velocity dispersion
• Surface acoustic waves
– Penetration depth ~λ
– Multilayers: dispersion
coating
substrate
substrate
coating
Dispersion curve:
Laser ultrasonics on layered samples: elastic information from guided wave velocity dispersion
Rayleigh waves: wavelength dependent penetration depth
0.1 1 10 100 1000
2000
3000
4000
5000
6000
7000
8000
9000
cT,Sn = 1586 m.s-1 cR,Sn = 1485 m.s-1
cL,Si = 8430 m.s-1
cR,Si = 5148 m.s-1
frequency (MHz)
velo
city
(m.s
-1)
Sn
Cu
Si
CuxSny
Photoacoustic characterization of elastic properties of (sub-)micron sub-surface layers
Multilayer sample
0
1
2
0
1
2
P (a
.u.)
0 2 4 6 8 100
1
2
3
z (µm)RayleighSezawa
5 MHz
50 MHz
240 MHz
0.1 1 10 100 1000
2000
3000
4000
5000
6000
7000
8000
9000
cT,Sn = 1586 m.s-1 cR,Sn = 1485 m.s-1
cL,Si = 8430 m.s-1
cR,Si = 5148 m.s-1
frequency (MHz)
velo
city
(m.s
-1)
Sn
Cu
Si
CuxSny
Photoacoustic characterization of elastic properties of (sub-)micron sub-surface layers
Rayleigh waves: wavelength dependent penetration depth
Photoacoustic characterization of elastic properties of free-standing films and plates: extraction of velocity dispersion
0
100
50
0.5
1000 806040200
1
10 20 30 40 50 60 70 80 90 100
20
40
60
80
100
0 10 20 30 40 50 60 70 80 90 100
-5
0
5
( , ) ( )
( ) exp( )
with =
( )
kc
= ( , ) exp( ) 2
S x t S x ct
S k i t ikx dk
ddk S k i t ikk xc
ω
ω
ω ω ωδ ωπ
+∞
−∞
+∞ +∞
−∞ −∞
= −
= −
−−
∫
∫ ∫
Photoacoustic characterization of elastic properties of free-standing films and plates: extraction of velocity dispersion
x
t
t
prob
e-pu
mp
dist
ance
frequency (Hz) 10
610
710
810
9
2050
2100
2150
2200
phas
e ve
loci
ty (m
/s)
k
ω2D Fourier transform
c=ω/k
( , ) ( ) exp( ) with = k c( )
( ( ( ), ) exp( ) 2
)
S x t S k i t ikx dk k
ddk S k c i t ikxk
ω ω
ω ω ωπ
δ ω ω
+∞
−∞
+∞ +∞
−∞ −∞
= −
= − −
∫
∫ ∫
substrate
coating
Photothermal characterization of thermal properties of free-standing films and plates: extraction of effective thermal diffusivity dispersion
Optical detection schemes
Laser ultrasonics ⇒ detection schemes
detection tecnhiques
optical
laser beam deflection
knife edge detection
speckle knife edge detection
(SKED)
diffraction interferometric
no scattering
Michelson common-path
Sagnac
birefringent crystal
Phase mask
scattering
laser Doppler
speckle
shearography
photorefractive
optical fiber
modulated optical
reflection
Brillouin oscillations
piezoelectric
Deflection angle ∆θ ≅ ξ/w ~ spatial derivative of wave packet ~ high pass response
w = characteristic lateral dimension of wave packet= displacement ξ or acoustic wavelength λacoustic
Deflection displacement: δ=∆θRelative differential intensity:∆I/I=δ/φ2φ2 is the diameter of the reflected beam at lens L2φ2=F2/F1φ1φ1 is the diameter of the incoming beam at lens L1
So that ∆I/I=δ/φ2=∆θ F1/φ1=(ξ/w)(F1/φ1)
E.g.w=10 µm, F1=50mm,φ1=5mmDisplacement detection limit:ξmin=(∆I/I)min (F1/φ1)-1 w=10-10w=10-15 m(W/Hz)1/2
E.g. 300Hz bandwidth, 75µW probe laser power:Typical light intensity change detection limit:10-9(W/Hz)1/2
ξmin,typ=2pm
2
20
222
2
2 exp2
exp
x dxdI OI x dx
ξ
φ δ ξφφ π
φ
+∞
−∞
− = ≅ = −
∫
∫
F2
F1
Z
Detectors
δ
θ
Laser ultrasonic detection schemes: laser beam deflection
wξ
Z
F2
F1
Detectors
t
difference signal
Laser ultrasonic detection schemes: laser beam deflection
Z
F2
F1
Detectors
t
difference signal
Z
F2
F1
Detectors
t
difference signal
Z
F2
F1
Detectors
t
difference signal
Z
F2
F1
Detectors
t
difference signal
F2
F1
X
Z
Detectors
Pixel signal sign assignment based on steady state light pattern32 x 32 photodiode pixel array
Laser ultrasonic detection schemes: speckle knife edge detector (SKED)
1 2 1 2( ) 2 cos( )I t I I I I φ= + + ∆PHOTO
DETECTOR
NON-CONTACT FAST SENSITIVE
2 ξφ πλ
∆ =λ/4
displacement δ(t)optical phase ∆φ
light intensity
Laser ultrasonic detection schemes: Michelson interferometer
PD
Fig. 2a. Grating Interferometer Transmission Mode
Fig. 2b. Grating Interferometer Reflection Mode
Pm1 L1 L2 Pm2
PC
Probe
Pump
Probe
PD
Pump
Sample
OptionalSample
L/2 L/4
Al-coatedsubstrate
CpL
Pm
Transmission mode
Reflection mode
Laser ultrasonic detection schemes: phase mask interferometer
http://kino-ap.eng.hokudai.ac.jp/interferometer.html
Laser ultrasonic detection schemes: common path interferometer: Sagnac configuration
Laser ultrasonic detection schemes: common path interferometer: time delay by birefringent crystal
Laser ultrasonic detection schemes: laser Doppler vibrometry
• 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐸𝐸𝑟𝑟 𝑡𝑡 ² + 𝐸𝐸𝑠𝑠 𝑡𝑡 ² + 2 ∗ 𝐸𝐸𝑟𝑟 𝑡𝑡 ∗ 𝐸𝐸𝑠𝑠 𝑡𝑡 ∗ cos(𝜑𝜑)
• with 𝜑𝜑 = 𝑘𝑘 ∗ ∆𝑥𝑥 = 2𝜋𝜋∆𝑥𝑥/𝜆𝜆
• 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐸𝐸𝑟𝑟 𝑡𝑡 ² + 𝐸𝐸𝑠𝑠 𝑡𝑡 ² + 2 ∗ 𝐸𝐸𝑟𝑟 𝑡𝑡 ∗ 𝐸𝐸𝑠𝑠 𝑡𝑡 ∗ cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆
• 𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐼𝐼𝑟𝑟 + 𝐼𝐼𝑠𝑠 + 2 ∗ 𝐼𝐼1𝐼𝐼2cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆
cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆
= cos 𝜔𝜔𝑟𝑟 − 𝜔𝜔𝑠𝑠 ∗ 𝑡𝑡 = cos 𝜔𝜔0 + 𝜔𝜔𝐴𝐴𝐴𝐴𝐴𝐴 − 𝜔𝜔𝑡𝑡 − ∆𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑡𝑡
cos 2𝜋𝜋 𝑟𝑟𝑟𝑟−𝑟𝑟𝑠𝑠𝜆𝜆
= cos 𝜔𝜔𝐴𝐴𝐴𝐴𝐴𝐴 − ∆𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑡𝑡
𝐼𝐼𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐼𝐼𝑟𝑟 + 𝐼𝐼𝑠𝑠 + 2 ∗ 𝐼𝐼1𝐼𝐼2 cos 𝜔𝜔𝐴𝐴𝐴𝐴𝐴𝐴 − ∆𝜔𝜔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑡𝑡 detected byphotodetector
( )0
( )( ) sample
sample AOM
v tt
cω ω ω∆ = +
Doppler effect
Phase locked loopPLL-detector
δ (m)
T
0 21
1 sin ( / 2)transmittedetalon
I IF φ
=+ ∆
0 0 2
2
0 2
11 sin ( / 2)sin ( / 2)
1 sin ( / 2)
reflectedetalon
etalon
etalon
I I IF
FIF
φ
φφ
= −+ ∆
∆=
+ ∆
δ (m)
R
R: mirror reflectivityF=4R/(1-R)2: Finesse∆φetalon=2π∆(ξ/λoptical)∆λoptical/λ0,optical=dξ/dt/c
Cavity resonance time:τcavity ~ Fdcavity/ce.g. dcavity=10mm, F=100:bandwidth ∆f=τcavity
-1=300MHz
cavity width dcavity
Laser ultrasonic detection schemes: Fabry-Perot interferometer
Laser ultrasonic detection schemes: broadband ←→ narrowband excitation
NarrowbandBroadband
λmin
Laser ultrasonic detection schemes: transient grating excitation
Heterodyne diffraction configuration
Laser ultrasonics:
material characterization
case studies
Laser ultrasonic applications: elastic characterization of hardened steel
excitation beam
bulk waves
thermal diffusion
← SAW SAW →
Laser excitation of surface acoustic waves (SAW) bulk waves thermal diffusion field
Optical detection
detection
excitation
Laser ultrasonic applications: elastic characterization of protective layer on fuel cladding
Laser ultrasonic applications: elastic characterization of protective layer on fuel claddingThermoelasticity GESA remelted 20 µm FeCrAlY on T91 steel
GESA sample
Laser ultrasonic applications: elastic characterization of protective layer on fuel cladding
Laser ultrasonic applications: elastic characterization of protective layer on fuel cladding
SAW velocity at different grating spacings
Laser ultrasonic applications: elastic characterization of protective layer on fuel cladding
Extracted coating thickness
from xy-scan from yx-scan
Laser ultrasonic applications: elastic characterization of protective layer on fuel cladding
Submicron oxide layer
d (µm) cR (m/s) ρ (kg/m³)
coating 0.6±1.0 (3.2±0.7) 10³ (7±11) 10³
substrate 2826 7990
Laser ultrasonic applications: elastic characterization of rough polymer coated steel sample
phasegrating
sample
L2 (sph.)
f1 f1+f2 f2
pumpprobe
differential photodetector
L1 (cyl.)phasegrating
sample
L2 (sph.)
f1 f1+f2 f2
pumpprobepumpprobe
differential photodetector
L1 (cyl.)
0 0.1 0.2 0.3 0.4 0.5
0
time (µs)
0 50 100 150 2000
0.05
frequency (MHz)
ampl
itude
(a
rb.u
nits
)
λ=29µm
50 100 150 200 300
0.1
1
wave
lengt
h (µ
m)
frequency (MHz)
wave
num
ber (
µm-1
)
93.5
43.529.121.814.6
9.56.6
k(f) polymer on steel
340 nm Al on polymer on steel
daverage,polymer=3µm
nanoindentation:
0 50 100 150 200 2501000
2000
3000
frequency (MHz)
velo
city
(m.s
-1)
(a)50 100 150
1000
2000
3000
frequency (MHz)
velo
city
(m.s
-1)
(b)
0 50 100 150 200 2501000
2000
3000
frequency (MHz)
velo
city
(m.s
-1)
(c)50 100 150
1000
2000
3000
frequency (MHz)
velo
city
(m.s
-1)
(d)
polymer on steel
2.4,3.3.6 GPa
0.15,0.30,0.35 GPa
Laser ultrasonic applications: elastic characterization of rubber layer
zxx
z
Laser ultrasonic applications: elastic characterization of rubber layer
2000
4000
6000
8000
velo
city
(m.s
-1)
longitudinal velocityCuxSny
shear velocityCuxSny
10 100
2000
4000
6000
8000
frequency (MHz)
velo
city
(m.s
-1)
thickness Sn
10 100frequency (MHz)
thickness CuxSny
Laser ultrasonic applications: elastic characterization of sub-micron intermetallic layer
Sn-Cu interdiffusion layer
Fitting parameters:
vL = 5400 m/svT = 2100 m/sd1 = 1.63 µmd2 = 5.10 µmd3 = 0.50 µmρ = 7900 kg/m³
10 100
2000
4000
6000
8000
10000
frequency (MHz)
velo
city
(m.s
-1)
4500 5000 5500 6000 6500 70000
0.5
1
1.5
2
vL CuxSny (m.s-1)
χ2 / χ2 m
in2050 2100 2150 2200
0
0.5
1
1.5
2
vT CuxSn
y (m.s-1)
χ2 / χ2 m
in
1 1.5 20
0.5
1
1.5
2
dSn (µm)
χ2 / χ2 m
in
4.8 5 5.2 5.40
0.5
1
1.5
2
dCu Sn (µm)
χ2 / χ2 m
in
Laser ultrasonic applications: elastic depth profiling of functionally graded materials
Experimental resultparameter study
38
E = 770 ± 26 GPaρ = 3250 ± 50 kg.m-3
L = 1.4 ± 0.04 µm
Laser ultrasonic applications: elastic characterization of sub-micron nanocrystalline diamond layer
39
Laser ultrasonic applications: thermal characterization of sub-micron nanocrystalline diamond layer
Laser ultrasonic applications: elastic characterization of Ni2MnGa layer
Calculation of guided wave dispersion and photothermal and photoacoustic
displacements
Bulk wave propagation in solidsPropagating quantities:• density ρ• displacement vector ui• strain tensor εij= ½(∂ui/∂xj+∂uj/∂xi)≡½(ui,j+uj,i)• stress components σij• velocity vector vi
2,
0 0 ,2ij ji
i ij jj
uut x
σρ ρ σ
∂∂≡ = ≡
∂ ∂
Governing equations:
• Newton
• Hooke
Strain
Stress
( )( )( ) ( )( )
2 2 20
2 2 20 , , ,
2 2
2
ij ijkl kl
ij L T kk ij T ij
L T k k ij T i j j i
c
c c c
c c u c u u
σ ε
σ ρ ε δ ε
ρ δ
=
= − +
= − + +
Bulk wave propagation in solids
2,
0 0 ,2ij ji
i ij jj
uut x
σρ ρ σ
∂∂≡ = ≡
∂ ∂
Combining Newton and Hooke:
( )( )( ) ( )( )
2 2 20
2 2 20 , , ,
2 2
2
ij ijkl kl
ij L T kk ij T ij
L T k k ij T i j j i
c
c c c
c c u c u u
σ ε
σ ρ ε δ ε
ρ δ
=
= − +
= − + +
( ) ( )( )( )
2 2 20 0 , , ,
2 2 2, ,
2
2
i L T k kj ij T i jj j ii
i L T j ji T i jj
u c c u c u u
u c c u c u
ρ ρ δ= − + +
= − +
Bulk wave propagation in solids: harmonic solutions in 2D
Combining Newton and Hooke:
( , , ) ( , ) exp( )( , , ) ( , ) exp( )x z t x z i t ikxx z t x z i t ikx
ϕ ϕ ωψ ψ ω
= −= −
2 2 22
2 2 2
2 2 22
2 2 2
L
T
kz c t
kz c t
ϕ ω ϕϕ
ψ ω ψψ
∂ ∂− + = −
∂ ∂
∂ ∂− + = −
∂ ∂
2 2 22 2
2 2 2
2 2 22 2
2 2 2
LL
TT
k pz c t
k pz c t
ϕ ω ϕ ϕ
ψ ω ψ ψ
∂ ∂= − ≡ ∂ ∂ ∂ ∂
= − ≡ ∂ ∂
( )( )
( , , ) exp( ) exp( ) exp( )
( , , ) exp( ) exp( ) exp( )L L
T T
x z t A p z B p z i t ikx
x z t C p z B p z i t ikx
ϕ ω
ψ ω
= + − −
= + − −
Harmonic proposal solution for plane waves running in the positive x-direction
Harmonic plane waves running in the positive x-direction with a depth profile in the z-direction.
Surface wave propagation in semi-infinite solids
( )( )
( , , ) exp( ) exp( ) exp( )
( , , ) exp( ) exp( ) exp( )L L
T T
x z t A p z B p z i t ikx
x z t C p z D p z i t ikx
ϕ ω
ψ ω
= + − −
= + − −
Harmonic plane wave components running in the positive x-direction with a depth profile in the z-direction.
Surface waves: no energy far away from the surface
( , , ) exp( )exp( )( , , ) exp( )exp( )
L
T
x z t B p z i t ikxx z t D p z i t ikx
ϕ ωψ ω
= − −= − −
Surface waves: no normal and no shear stress at the free surface: σzz=0 and σxz=0
( ) ( )( )2 2 20 , , ,2ij L T k k ij T i j j ic c u c u uσ ρ δ= − + +
=0
=0
2 2
2 2
20
2T T
L T
Bk p ikpDikp k p
+ − = +
Rayleigh determinant of homogeneous set of equations should be zero
Implicit condition relation between wave number k and angular frequency ω:dispersion relation
2 1/2 1/22 2 2 24
2 2 2 2 2 2 2 21 2 1 1 4 1 1 0T T T L
kk c k c k c k cω ω ω ω
+ + + + − + + =
4 2 2 4 22 4 0T T T Lk k p p k p p+ + − =
cR≡ω/k=f(cL,cT,ρ,ω,k)
NO DISPERSION
Guided wave propagation in a free-standing platewith thickness L
( )( )
( , , ) exp( ) exp( ) exp( )
( , , ) exp( ) exp( ) exp( )L L
T T
x z t A p z B p z i t ikx
x z t C p z D p z i t ikx
ϕ ω
ψ ω
= + − −
= + − −
Harmonic plane wave components running in the positive x-direction with a depth profile in the z-direction.
4 unknowns A,B,C,D to be found from 4 boundary condition equations:No normal and shear stress at plate boundaries z=0 and z=L:σzz(z=0)=0σzx(z=0)=0σzz(z=L)=0σzx(z=L)=0
( ) ( )( )2 2 20 , , ,2ij L T k k ij T i j j ic c u c u uσ ρ δ= − + +
0
ACBD
=
f(cL,cT,ρ,ω,k,L)=0
Lamb determinant of homogeneous set of equations should be zero
cLamb(cL,cT,ρ,ω,L)
DISPERSION
MULTIPLE SOLUTIONS PER FREQUENCY
Zero search of determinant of set of equations, looking for root k-values for every value of ω
Guided wave propagation in a free-standing platewith thickness L
f(cL,cT,ρ,ω,k,L)=0
S0-mode A0-mode
Aluminium plate L = 2 mm, ρ = 2700 kg/m³, vT = 3040 m/s, vL =6420 m/s, vR = 2846 m/s
zxx
z
• Dispersion• Multiple modes• Both
running (Im(k)=0) andlocal modes (Im(k)≠0)
Guided wave propagation in multilayers
( )( )
1 1
1 1
1 1 1
1 1 1
( , , ) exp( ) exp( ) exp( )
( , , ) exp( ) exp( ) exp( )
L L
T T
x z t A p z B p z i t ikx
x z t C p z D p z i t ikx
ϕ ω
ψ ω
= + − −
= + − −
( )( )
2 2
2 2
2 2 2
2 2 2
( , , ) exp( ) exp( ) exp( )
( , , ) exp( ) exp( ) exp( )
L L
T T
x z t A p z B p z i t ikx
x z t C p z D p z i t ikx
ϕ ω
ψ ω
= + − −
= + − −
( )( )
3 3
3 3
3 3 3
3 3 3
( , , ) exp( ) exp( ) exp( )
( , , ) exp( ) exp( ) exp( )
L L
T T
x z t A p z B p z i t ikx
x z t C p z D p z i t ikx
ϕ ω
ψ ω
= + − −
= + − −
0
1
2
3
layer
σzz(z=0)=0σzx(z=0)=0
σzz+(z=d1)=σzz
- (z=d1)σzx
+ (z=d1)=σzx- (z=d1)
uz+ (z=d1)=uz
- (z=d1)ux
+ (z=d1)=ux- (z=d1)
σzz+(z= d1+d2)=σzz
- (z= d1+d2)σzx
+ (z= d1+d2)=σzx- (z= d1+d2)
uz+ (z= d1+d2)=uz
- (z= d1+d2)ux
+ (z=d1+d2)=ux- (z= d1+d2)
A3=0C3=0
12 unknownsA1,B1,C1,D1A2,B2,C2,D2A3,B3,C3,D3
12 equations
What if there is a delamination?
1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 1,10 1,11 1,12 1
2,1 1
3,1 1
4,1 1
5,1 2
6,1 2
7,1 2
8,1 2
9,1 3
10,1 3
11,1 3
12,1 12,12 3
...
M M M M M M M M M M M M AM BM CM DM AM BM CM DM AM BM CM M D
0=
What if there is a source?
f(cL,cT,ρ,ω,k,L)=0
Determinant of homogeneous set of equations should be zero
cLamb(cL,cT,ρ,ω,L)
DISPERSION CURVE SOLUTIONS k(ω), c(ω)
Zero search of determinant of set of equations, looking for root k-values for every value of ω
f(material properties,ω,k,L)=0
Rayleigh waves: wavelength dependent penetration depth
0.1 1 10 100 1000
2000
3000
4000
5000
6000
7000
8000
9000
cT,Sn = 1586 m.s-1 cR,Sn = 1485 m.s-1
cL,Si = 8430 m.s-1
cR,Si = 5148 m.s-1
frequency (MHz)
velo
city
(m.s
-1)
Sn
Cu
Si
CuxSny
Multilayer sample
Guided wave propagation in multilayers
50
2
2
T C T Qtx
ρκ κ
∂ ∂− = −
∂∂( )
02
cos( )( , )2
Q kxT xk i C
ωπ κ ω
=+
0( , ) exp( )Q x t Q i t ikxω= + ( ) ( )2
02 2 2 2
sin( )( , )2
L
L
k c Q qxu xC k i k c
γωπρ α ω ω
−=
+ −2 2
2 2 2
1
L
u u Txx c t
γ∂ ∂ ∂− =
∂∂ ∂
0
0
( , ) ( ) cos( )( , ) cos( )
Q x t Q t kxQ x Q kx
δω=
⇒ =
cL: longitudinal speed of soundC: specific heat capacityρ: densityκ: thermal conductivityγ: thermal expansion coefficientα = κ/ρC : thermal diffusivity
log(t)
dens
ity g
ratin
gam
plitu
deSolution of longitudinal wave equation without and with a source
WITH A THERMAL SOURCE
( )2 0
( , ) exp( ) cos( )2
exp( ) cos
(
( )2
( )
12
) dT x t i t kx
di
QS
k i Ct kQ x
ωωπ
ω
π κω
ω
ωπω
+∞
−∞
+∞
−∞
=
=+
∫
∫
determinant!
Residue theorem?
Kramers rule?
51
2
2
T C T Qtx
ρκ κ
∂ ∂− = −
∂∂( )
02
cos( )( , )2
Q kxT xk i C
ωπ κ ω
=+
0( , ) exp( )Q x t Q i t ikxω= + ( )( )2
02 2 2 2
sin( )( , )2
L
L
k c Q kxu xC k i k c
γωπρ α ω ω
−=
+ −2 2
2 2 2
1
L
u u Txx c t
γ∂ ∂ ∂− =
∂∂ ∂
0
0
( , ) ( )sin( )( , ) sin( )
Q x t Q t kxQ x Q kx
δω=
⇒ =
cL: longitudinal speed of soundC: specific heat capacityρ: densityκ: thermal conductivityγ: thermal expansion coefficientα = κ/ρC : thermal diffusivity
log(t)
dens
ity g
ratin
gam
plitu
deSolution of longitudinal wave equation without and with a source
WITH A THERMAL SOURCE
( )( )2
2 2 2 2 0
( , ) exp( ) sin( )2
exp
( )
( ) s2
(
)2
)
in(L
L
du S
k cC k i
x t i t kx
di t kx
Q
k cQ
ωωπ
ωωπ
ω
γπρ α ω
ω
ω
+∞
−∞
+∞
−∞ −
=
=−+
∫
∫
determinant!
Residue theorem?
52
2
2
T C T Qtx
ρκ κ
∂ ∂− = −
∂∂( )
02
cos( )( , )2
Q kxT xk i C
ωπ κ ω
=+
0( , ) exp( )Q x t Q i t ikxω= + ( )( )2
02 2 2 2
sin( )( , )2
L
L
k c Q kxu xC k i k c
γωπρ α ω ω
−=
+ −2 2
2 2 2
1
L
u u Txx c t
γ∂ ∂ ∂− =
∂∂ ∂
0
0
( , ) ( )sin( )( , ) sin( )
Q x t Q t kxQ x Q kx
δω=
⇒ =
( )202 2
2
sin( )( , ) exp cos( ) sin( ) ( )1
L LL
L
Q kx ku x t k t kc t kc t tckCk
c
γ αα θαρ
= − − + +
( )20 sin( )( , ) exp ( )Q kxT x t k t tC
α θρ
= −
cL: longitudinal speed of soundC: specific heat capacityρ: densityκ: thermal conductivityγ: thermal expansion coefficientα = κ/ρC : thermal diffusivity
log(t)
dens
ity g
ratin
gam
plitu
deSolution of longitudinal wave equation without and with a source
WITH A THERMAL SOURCE
determinant!
Thermoelastic excitation
Proposal solutions
Potentials wave equation
with
Displacements at the surface
with
with
with
k,ω domain thermal driving source
Transient grating in reflection mode: displacement response
2 remaining unknowns B and D are determined via 2 boundary conditions
Stresses are determined by Duhamel-Neumann relation
A = 0C = 0 effect of surface tension
effect of gravitation thermal driving source
Transient grating in reflection mode: displacement response
Unknowns A,C,E
~ source
effect of surface tension quasi-Rayleigh determinant
2x2 set of equations in unknown coefficients B and D
Solution:
Transient grating in reflection mode: displacement response
Solution for arbitrary source = 2D (k→x,ω→t) Fourier transform of solution for harmonic excitation, with source spectrum I(ω,k) as weighting function
Transient grating in reflection mode: displacement response:arbitrary source
( )
( ) ( )00
00
( , , ) ( , ) exp ( )
( , )( , , ) ( , ) exp( ( , ) ) ( , ) exp( ( , ) ) ( , ) exp( ( , ) ) exp
( , )( , , ) ( , ) exp( ( , ) ) ( , ) exp(
L L
T T
I x z t d dk I k ikx i t z
I kx z t d dk A k p k z B k p k z E k k z ikx i tI
I kx z t d dk C k p k z D k pI
ω ω ω δ
ωφ ω ω ω ω ω ω σ ω ω
ωψ ω ω ω ω
+∞ +∞
−∞ −∞
+∞ +∞
−∞ −∞
= +
= + − + − +
= + −
∫ ∫
∫ ∫
( ) ( )( , ) ) expk z ikx i tω ω+∞ +∞
−∞ −∞
+∫ ∫
( ) ( )
( , , ) ( ) ( ) ( ) ( , ) 1
( , , 0) ( , ) ( , ) ( , ) ( , ) ( , ) exp z L
I x z t x z t I k
u x t z d dk p k B k k E k ikD k ikx i t
δ δ δ ω
ω ω ω σ ω ω ω ω+∞ +∞
−∞ −∞
= ⇒ =
⇓
= = − − + +∫ ∫
( ) ( )
( ) ( )
( )
0 00 0
0 0
0
0
0
( , , 0)
exp( ) exp( ) exp( ) exp
( , )
( ,
z z zz z
z
L L T
z
L
I kI
u k z
z x
p B p z E z ikD p z ikx i t
p B E ikD I k
ω φ ψ
φ ψ
σ σ ωω
ωσ
= =
=
=
= = ∇ + ∇×
∂ ∂ = + ∂ ∂
= − − − − + − +
= − − + ( )0
exp ) ikx i tI
ω+
Poles!Residue theorem
Laser ultrasonics for defect detection and application for non-destructive testing
3 mm
Lamb waves on a thin Cu membrane …Typical result from full field Doppler xy scanning
Laser ultrasonic laser Doppler xy scanning imaging: example of material characterization and non-destructive testing
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