aeem-7028 lecture, part 6 attenuation - aerospacepnagy/classnotes/aeem7028 ultrasonic nde... ·...
TRANSCRIPT
Part 6
ATTENUATION
Signal Loss
Loss of signal amplitude:
1 1
2 2[Neper] ln or [dB] 20logA AL L
A A= =
A1 is the amplitude without loss
A2 is the amplitude with loss
Proportional loss of signal amplitude with increasing propagation distance:
L d= α
d is the propagation distance
α is the attenuation coefficient
Major classes of attenuation:
absorption (viscosity, relaxation, heat conduction, elastic hysteresis, etc)
scattering (inhomogeneities), also causes incoherent material noise
absorption scatteringα = α + α
For example: 2water [dB/m] 0.2 [MHz]fα ≈
Plexiglas [dB/m] 100 [MHz]fα ≈
f denotes frequency
Scattering Induced Attenuation
Single-scatterer:
sP I= γ
Ps scattered power by the inhomogeneity
γ is the scattering cross section of the average scatterer
I intensity of the incident wave
Single-scattering approximation:
Let us consider a volume of given cross section A and length d. The coherent acoustic power P A I= transmitted through this region decreases by an amount of
s s1
Ni
idP P P N I n Ad I n d P
=− = = = γ = γ = γ∑
N number of scatterers in volume Ad
n the number density of the scatterers
dP P n d= − γ
i n dP P e− γ=
½i n du u e− γ=
½ nα = γ
0lim 0ω→
γ =
slim 2 Aω→∞
γ ≤
General Considerations on Scattering
Similarity:
d1
D1
λf ,1 1
d2
D2
λf ,2 2
Scaling:
2 2 2 1
1 1 1 2
d D fd D f
⎛ ⎞λξ = = = =⎜ ⎟λ ⎝ ⎠
Scattering Loss:
1 2L L=
1 1 1
2 2 2
//
L dL d
α= = ξ
α
1 1
2 21α λ
=α λ
Normalized Attenuation:
n n( , ) ( )DDα = λα λ = αλ
Power relationship:
n ( , ) i iD f D fα ∝
1( , ) i iD f a D f +α =
a is a constant determined mainly by the "degree" (relative deviation from the host medium) of
the inhomogeneity and the nature of the interaction (e. g., shear or longitudinal wave, etc.)
Polycrystalline material:
D is the grain size
a is a function of anisotropy
Low-frequency (Rayleigh) region:
3 4Rayleigh R( , )D f a D fα =
Intermediate (stochastic) region:
2stochastic s( , )D f a D fα =
High-frequency (geometrical) region:
1geometrical g( , )D f a D−α =
Surface wave attenuation on a slightly rough surface:
4 5roughness r( , )h f a h fα =
h is the rms roughness
ar is a function of the rms roughness-to-autocorrelation length ratio
Scattering Induced Attenuation in Polycrystalline Materials
Low-Frequency (Rayleigh) Region ( Dλ >> )
r
Scattered Wave
Scatterer
Incident Wave
uiu s
( )s s0( , ) ( )
i k r teu r u Fr
− ωθ = θ
s0u r/ is the amplitude of the wave at a distance r from the scatterer
( )F θ is the directivity function (θ denotes the polar angle)
Linear superposition:
s0 iu V u∝ Δ
iu is the incident wave amplitude
Δ is the relative change of the elastic properties
V is the scatterer volume
For example, for cubic crystals:
44
11 12
2 1CC C
Δ = −−
The total scattered power from a single scatterer:
2 2 2i2 2 2
s s i2S S
V uP u dS dS V I
rΔ
∝ ∝ ∝ Δ∫ ∫
2i iI u∝ denotes the intensity of the incident wave
½ nα = γ
2 2Vγ ∝ Δ is the scattering cross section
1n V −= and 3V D∝
2Rayleigh Vα ∝ Δ
3 4 2 3 4Rayleigh Ra D f D fα = ∝ Δ
Ra is a constant that is proportional to 2Δ
Intermediate (Stochastic) Region ( Dλ ≈ )
Incident Ray θdivergence
θrefraction
Refracted Ray
In the case of weak anisotropy:
refractionθ ≈ Δ
divergence Dθ ≈ λ/
Geometrical region:
divergence refractionθ ≤ θ
i. e., above a frequency where Dλ ≤ Δ/
Stochastic region (forward scattering):
weak random phase perturbation ( , )x yΦ multiplies the coherent (average) wave
2( , ) ( , ) ½ ( , )i x y i x y x ye e eΦ <Φ > − <Φ >< > ≈
Loss of the coherent wave:
2½L ≈ ϕ
2 2( , )x yϕ = < Φ >
D fϕ ∝ Δ
2 2 2L D f∝ Δ
2 2 2stochastic sa D f D fα = ∝ Δ
sa is a constant that is proportional to 2Δ
High-Frequency (Geometrical) Region ( Dλ << )
d
Incident Plane Wave
Transmitted Plane Wave
1geometrical D−α ∝
Summary:
Regime Functional Dependence
Rayleigh ( D << λ ) 2 3 4D fα ∝ Δ
stochastic ( D DΔ≤ λ ≤ ) 2 2D fα ∝ Δ
geometrical ( Dλ ≤ Δ ) 1D−α ∝
Grain Scattering Induced Attenuation In Polycrystalline Iron
(100 μm grain diameter)
log{Frequency [MHz]}
log{
Atte
nuat
ion
Coe
ffic
ient
[dB
/cm
]}
-5
-4
-3
-2
-1
0
1
2
3
-1 0 1 2 3
Rayleigh region stochastic region geometrical region
shear
longitudinal
Experimental Grain Scattering Induced Attenuation
longitudinal wave in SAE 1020 steel
Frequency [MHz]
Atte
nuat
ion
Coe
ffic
ient
[dB
/cm
]
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20
57 µm48 µm38 µm31 µm18 µm10 µm
Complicating effects:
preferred orientation between neighboring grains (e. g., prior austenite grain structure, columnar grain structure, severe plastic flow, etc.)
shape of the grains can also be very different from the ideal uniaxial shape (e. g., needle-like
alpha (hexagonal) grains in titanium)
Ultrasonic Grain Size Assessment Typical welded zone:
Over-heated welded zone:
transmission C-scan of an app. 4"-wide electric resistance welded butt joint between two 0.25"-thick steel plates at 15 MHz
electrode
grain coarsening
plastic flow
high-pressure side
low-pressure side
interface
Experimental Aspects of Ultrasonic Attenuation Measurements
time
“main bang” A0 A1 A2 • • •
time
“main bang” A0 A1 A2 • • •
0imp diff surf
120log 2AL d L L L
A= = α + + +
Impedance Mismatch:
R0R1 R2
R3
T1
T2T3
d
I
Medium 1
Medium 1 (3)
(sample)Medium 2
Reflection coefficient:
2 112
2 1
Z ZRZ Z
−=
+
12 21R R R= = −
2 212 21 1T T T R= = −
Z1 and Z2 are the acoustic impedances of the first and second media, respectively
Front surface reflection:
0R R=
Multiple-reflection:
21R R T= , 3 22R R T= , ... 2 1 2nnR R T−=
Transmission:
21T T= , 2 22T R T= , ... 2( 1) 2nnT R T−=
where n = 1, 2, ...
0 2imp
120log 40log 20log(1 )RL T R
R= = − = − −
Diffraction Correction
the acoustic field of a circular piston radiator at a /λ = 10
z = a
near-field
far-field
z = a = N10
z = a20
Near-field / far-field transition: 2aN =λ
a denotes the radius of the transducer λ is the acoustic wavelength in the medium
Simplified Sound Field of a Circular Piston Radiator
-2
-1
0
1
0 1 2 3 4
2-10dB contour"searchlight"
model
simplified model
θ-10 dB aξ
zN
Two identical transducers in a pitch-catch mode:
rL
r
( )( )( 0)p zD z
p z=
=
z is the distance between the transducers (in a pulse-echo operation with a normally aligned mirror, the distance between the transducer and the mirror is only z /2)
Lommel diffraction correction:
2L 0 1( ) 1 [ (2 ) (2 )]i sD s e J s i J s− π= − π + π/ / /
s = z /N In the far-field:
Llim ( )s
D s i s→∞
= π/
2Llim ( )
z
aD zz→∞
π=
λ
Lommel Diffraction Correction for a Circular Piston Radiator
z / N
Diff
ract
ion
Cor
rect
ion
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
far-field asymptote
circular piston radiator of 0.5"-diameter and z = 20 cm separation in water
Frequency [MHz]
Diff
ract
ion
Cor
rect
ion
[dB
]
-10
-8
-6
-4
-2
0
0 10 20 30 40 50
Refraction Correction Snell’s Law:
2 2
1 1
sinsin
cc
θ=
θ
fluid solid
For slightly divergent beams:
sin tanθ ≈ θ ≈ θ
21 2
1
cz z zc
= +
Pulse-echo configuration:
1L
1diff
1 2 1L
1
2
20log2 2 /
zDN
Lz d c cD
N
⎛ ⎞⎜ ⎟⎝ ⎠≈
⎛ ⎞+⎜ ⎟⎝ ⎠
2 2
11 1
a a fNc
= =λ
Surface Roughness
Rough Surface
Transducer
Flaw
Liquid
Solid
Incident Wave Coherent Reflection
z
x
θI θR
Incoherent Reflection
Incoherent Transmission
Coherent ShearTransmission
CoherentLongitudinalTransmission
θT
θL
Phase-Screen Approximation
( , )s x y is the surface height distribution
h is the rms height
Λ is the correlation length
2 2( , )h s x y= < >
2( , ) ( , ) ( , ) ( , )C s x y s x y h cξ η = < − ξ − η > = ξ η
transverse isotropy:
2 2 2ρ = ξ + η
Gaussian distribution:
2 2/( )c e−ρ Λρ =
Logarithmic distribution:
/( )c e−ρ Λρ =
small curvature: h << Λ
Phase perturbation (without the common e-iωt term)
( , , )0( , , , 0 ) ( , , , ) i x yx y z x y z e− Φ ωω = = ωu u
u denotes the displacement field just inside the rough solid
0u denotes the displacement field just inside the smooth solid
L,T w I L,T L,T( , ) [ cos( ) cos( )]s x y k kΦ = θ − θ
R w I2 ( , ) cos( )s x y kΦ = θ
kL, kT and kw are wavenumbers
“Coherent” Transmission Coefficients
2½L ≈ < Φ > Reflected compressional wave:
0R ( , ) 20log RL
Rω θ =
Longitudinal transmitted wave:
L0L
L( , ) 20log
TL
Tω θ =
Shear transmitted wave:
T0T
T( , ) 20log
TL
Tω θ =
Phase-screen approximation:
2 2R,L,T R,L,T = 8.686 [dB] L h Cω
2R I w= 2 [cos( ) ]C cθ /
1 2-L L L I w2 = [cos( ) cos( ) ]C c cθ θ/ /
1 2-T T T I w2 = [cos( ) cos( ) ]C c cθ θ/ /
cw, cL and cT are sound velocities
Surface Roughness Induced Attenuation of the Reflected Ultrasonic Wave at Normal Incidence
Frequency [MHz]
Atte
nuat
ion
[dB
]
0
5
10
15
20
25
30
35
40
0 5 10 15 20
45.6 µm25.6 µm15.2 µm12.8 µm11.4 µm9.9 µm
8.7 µm5.6 µm
(solid lines are best fitting f 2 curves)
Surface Roughness Induced Attenuation of the Double-Transmitted Longitudinal and Shear Waves
Frequency [MHz]
Atte
nuat
ion
[dB
]
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10 12 14 16 18 20
0º reflection 12º long. tr.10º long. tr.
0º long. tr.26º shear tr.24º shear tr.22º shear tr.
(solid lines are best fitting f 2 curves).