Ultrasonics

Fundamentals

Applications

Simple Harmonic Wave

x

λ

t = t1t2

t3

cu

u0

-u0

0( , ) cos[ ( ) ]x

u x t u tc

= ω − + ϕ

u displacement 0u denotes the amplitude 2 fω = π is the angular frequency

f is the cyclic frequency ϕ is the phase angle at 0x t= =

c denotes the propagation (phase) velocity

( )0( , ) i k x tu x t U e± − ω=

0U is a complex amplitude 2 /k = π λ is the wave number

λ is the wavelength

0 cos( )xu u e k x t−α= − ω − ϕ

α is an attenuation coefficient

Standing Wave

0 0cos( ) cos( )u u k x t u k x t= + ω + − ω 02 cos( ) cos( )u k x t= ω

Successive instants of standing wave vibration in a specimen.

x

λ

t = t1t2

t3

unode

antinode

u0

-2u0

2

A node is a point, line, or surface of a vibrating body that is free from vibratory motion.

Arbitrary Pulse and Harmonic Wave Packet

u

x

c

u

x

c

f ( x - c t )

cos [ k ( x - c t ) ]

f ( x - c t ) f ( x - c [ t + dt ] )

Pulse of arbitrary shape

( )u f x c t= −

Oscillatory wave packet

( )cos[ ( )]u f x c t k x c t= − −

Fundamental Wave Modes Longitudinal Wave:

wavedirection

Shear Wave:

wavedirection

Surface Wave:

wavedirection

Acoustic Wave Interaction with Material Discontinuities

ρ , c1 1

ρ , c2 2

Incident Wave Reflection

Transmission

Liquid

Solid

Incident Wave Reflection

θi θr

ShearTransmission

LongitudinalTransmission

θs

θd

Incident Wave Reflection

θi θr

Edge Diffraction

Longitudinal Wave Propagation in Thin Rods

dxx

u

dx

σ ∂σσ + dx∂x

Equation of motion:

2

2( )

udx A A Adx

x t

∂ σ ∂σ + − σ = ρ∂ ∂

or 2

2u

x t

∂ σ ∂= ρ∂ ∂

where A is the cross-sectional area and ρ is the mass density. Constitutive equation:

Eσ = ε

where ε is the axial strain in the material and E denotes Young's modulus. Displacement-strain relationship:

u

x

∂ε =∂

Wave equation:

2 2

2 2u u

Ex t

∂ ∂= ρ∂ ∂

or 2 2

2 2 21

rod

u u

x tc

∂ ∂=∂ ∂

, where rodE

c =ρ

Solution of the Wave Equation

2 2

2 2 21u u

x c t

∂ ∂=∂ ∂

where c is the wave velocity:

stiffnessvelocity

density=

Propagating harmonic wave represents a solution of the wave equation:

0( , ) cos[ ( ) ]x

u x t u tc

= ω − + ϕ

Arbitrary wave pulse of the general form ( , ) ( )x

u x t f tc

= − also satisfies the wave

equation:

2

2( , ) ''( )

xu x t f t

ct

∂ = −∂

2

2 21

( , ) ''( )x

u x t f tcx c

∂ = −∂

Dilatational Modes

stiffnessvelocity

density=

Thin Rods:

, 0x x y zEσ = ε σ = σ = ⇒ rodE

c =ρ

wavedirection

Thin plates:

21

x xEσ = ε

− ν, 0y zε = σ = ⇒

2 21.05 (for 0.3)

(1 ) 1

rodplate rod

cEc c= = ≈ ν =

− ν ρ − ν

Infinite Medium:

(1 )

(1 )(1 2 )x xE − νσ = ε

+ ν − ν, 0y zε = ε = ⇒

(1 ) (1 )1.16

(1 ) (1 2 ) (1 ) (1 2 )d rod rodE

c c c− ν − ν= = ≈

+ ν − ν ρ + ν − ν

Transverse (Shear) Waves longitudinal transverse (dilatational, compressional) (shear)

x

σx

σy

σx-

σy-

y

ux

-σ

x

σyx

yx

σxyσxy-

uy

, yxy xy xy

u

x

∂σ = µ γ σ =

∂

2 2

2 2 21y y

s

u u

x tc

∂ ∂=

∂ ∂

scµ=ρ

2 2

1 2d

s

c

c

− ν=− ν

Acoustic Impedance The relationship between stress σ, displacement u, and particle velocity v for a propagating wave is of interest. As an example, let us consider a dilatational wave propagating in an infinite elastic medium:

( )( , ) i k x txu x t Ae − ω=

( )( , ) x i k x tx

ux t i Ae

t− ω∂

= = − ω∂

v

( )x i kx tx xx xx

uC C Ai k e

x− ω∂= =σ

∂

The ratio of the pressure (or negative stress) to the particle velocity is called the acoustic impedance. For a dilatational wave propagating in the positive direction,

2 ( )

( )

i kx tx d

d di k x tx

c Ai k eZ c

i Ae

− ω

− ωρσ= − = = ρωv

The product of density and wave velocity occurs repeatedly in acoustics and ultrasonics and is called the characteristic acoustic impedance (for a plane wave). It is the impedance that acoustically differentiates materials, in addition to the moduli and density.

Densities, Acoustic Velocities and Acoustic Impedances of Some Materials

Material Density, [103 kg/m3]

ρ

Acoustic velocities [103 m/s]

long. dc shear sc

Acoustic impedance

[106 kg/m2s]

dZ

Metals

Aluminum 2.7 6.32 3.08 17 Iron (steel) 7.85 5.90 3.23 46.5 Copper 8.9 4.7 2.26 42 Brass 8.55 3.83 2.05 33 Nickel 8.9 5.63 2.96 50 Tungsten 19.3 5.46 2.62 105 Nonmetals

Araldit Resin 1.25 2.6 1.1 3.3 Aluminum oxide 3.8 10 38 Glass, crown 2.5 5.66 3.42 14 Perspex (Plexiglas) 1.18 2.73 1.43 3.2 Polystyrene 1.05 2.67 2.8 Fused Quartz 2.2 5.93 3.75 13 Rubber, vulcanized 1.4 2.3 3.2 Teflon 2.2 1.35 3.0 Liquids

Glycerine 1.26 1.92 2.4 Water (at 20oC) 1.0 1.483 1.5

Reflection and Transmission at Normal Incidence

ρ , c1 1

ρ , c2 2

Incident Wave Reflection

Transmission

1cos( )i iu A k x t= − ω

1cos( )r ru A k x t= − − ω

2cos( )t tu A k x t= − ω

Boundary conditions: the displacements and stresses must be the continuous at the interface

i r tu u u+ = and i r tσ + σ = σ

1 21 2

1 21 2

rd

i

c cARc cA

− ρρ= =

+ρ ρ and 1 1

1 1 2 2

2td

i

A cT

A c c

ρ= =ρ + ρ

2 2 1 1

1 1 2 2

rs

i

c cR

c c

σ ρ − ρ= =σ ρ + ρ

and 2 2

1 1 2 2

2ts

i

cT

c c

σ ρ= =σ ρ + ρ

where R and T are known as the reflection and transmission coefficients. It is seen that these results are in terms of the respective acoustic impedances of the materials.

Example

reflected and transmitted (stress) amplitudes

pi

pr

pt

pt

pr

pi

steel water

water steel

Conservation of energy: the time rate of energy flow per unit area (i. e., intensity)

I p v v= = − σ

r t iI I I− + =

1 2 1 2 1 2

1 2 1 2 1 2 1 2

2 21d s d s

Z Z Z Z Z ZR R T T

Z Z Z Z Z Z Z Z

− −− + = + =+ + + +

Free surface ( 2 0Z → ):

1dR = , 1sR = − , 2dT = , 0sT =

Rigidly clamped surface ( 2Z → ∞ ):

1dR = − , 1sR = , 0dT = , 2sT =

Reflection and Transmission at Oblique Incidence

Mode Conversion

θdi

solid 1

Id Rd

Rs

Td

solid 2

Ts

z

y

θs1

θd1

θs2

θd2

solid 1

Rd

Rs

Td

solid 2

Ts

θsiIs

z

y

θs1

θd1

θs2

θd2

Boundary conditions: the displacements ( andy zu u ) and stresses ( andyy yzσ σ )

must be the continuous at the interface

Snell's Law:

1 1 2 2

1 1 1 1 2 2

sin sin sin sin sin sindi si d s d s

d s d s d sc c c c c c

θ θ θ θ θ θ= = = = =

General Solution

Constitutive equations:

( 2 ) yzyy

uu

z y

∂∂σ = λ + λ + µ∂ ∂

( )y zyz

u u

z y

∂ ∂σ = µ +∂ ∂

where 2 21 1 1 1 11 1, 2 ,s dc cµ = ρ λ + µ = ρ 2 2

2 2 2 2 22 2, and 2s dc cµ = ρ λ + µ = ρ .

Boundary conditions:

(2) (1) ( 1) ( 2) ( 1) ( 2)

(2) (1) ( 1) ( 2) ( 1) ( 2)

(2) (1) ( 1) ( 2) ( 1) ( 2)

(2) (1) ( 1) ( 2) ( 1)

0

0or

0

0

d d s sy y y y y y

d d s sz z z z z z

d d s syy yy yy yy yy yy

d d szy zy zy zy zy z

u u u u u u

u u u u u u

− − + − + − − + − + = τ − τ − τ + τ − τ + τ

τ − τ − τ + τ − τ + τ

( )

( )

( )

( 2) ( )

iy

iziyy

s iy zy

u

u

= τ τ

11 12 13 14 1 1

21 22 23 24 2 2

31 32 33 34 3 3

41 42 43 44 4 4

or

d

d

s

s

a a a a R b c

a a a a T b c

a a a a R b c

a a a a T b c

=

depending on whether longitudinal or shear wave incidence is considered. aij , bi, and ci can be easily calculated from simple geometrical considerations.

(1) (2) (3) (4)det[ ] det[ ] det[ ] det[ ], , ,

det[ ] det[ ] det[ ] det[ ]d d s sR T R T= = = =a a a aa a a a

where ( )ia is the matrix obtained by replacing the ith column of a by either b or c vectors depending on whether longitudinal or shear incidence is used.

Energy Reflection and Transmission Coefficients

aluminum immersed in water

Angle of Incidence [deg]

Ene

rgy

Ref

lect

ion

& T

rans

mis

sion

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

reflection

longitudinaltransmission

shear transmission

steel immersed in water

Angle of Incidence [deg]

Ene

rgy

Ref

lect

ion

& T

rans

mis

sion

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

reflection

longitudinaltransmission

shear transmission

Wave Dispersion Dispersion means that the propagation velocity is frequency-dependent. Since the phase relation between the spectral components of a broadband signal varies with distance, the pulse-shape gets distorted and generally widens as the propagation length increases.

input pulse

ω∂c

> 0∂

ω∂c

= 0∂

ω∂c

< 0∂

Group Velocity

dispersive wave propagation of a relatively narrow band “tone-bursts”

phase velocity versus group velocity

phasevelocity

groupvelocity

Beating Between Two Harmonic Signals

1 1cos( )u t= ω

2 2cos( )u t= ω

1 2 1 21 2 1 2cos( ) cos( ) 2cos( ) cos( )

2 2u u t t t t

ω + ω ω − ω+ = ω + ω =

( , ) cos( ) cos[( ) ( ) ]

2cos( )cos( )2 2

u x t kx t k k x t

kk x t x t

= − ω + + δ − ω + δωδ δω≈ − ω −

where the first high-frequency term is called carrier wave and the second low-frequency term is the modulation envelope. This shows that the propagation velocity of the carrier is the phase velocity and the propagation velocity of the modulation envelope is the group velocity

ck

ω= gc

c c kk k

∂ ω ∂= = +∂ ∂

Material versus Geometrical Dispersion

Frequency [MHz]

Vel

ocity

[km

/s]

2.6

2.7

2.8

0 2 4 6 8 10

polyethylene

phase

group

lowest-order symmetric Lamb mode in a 1-mm-thick aluminum plate

Frequency [MHz]

Vel

ocity

[km

/s]

0

2

4

6

0 2 4 6

phase

group

Ultrasonic Transducers for NDE

Typical Acoustic Transducer Typical Ultrasonic Transducer

Electro-

TransformerMechanical

Mechanical-

TransformerAcoustical

V

Electro-

TransformerAcoustic

p p , σ, τV

< λ λ∼∼

>> Z Zt 0 Z Zt 0∼∼

Bandwidth, Pulse Length, and Axial Resolution

Am

plitu

de [a

. u.]

0 1 2 3

Spe

ctru

m [d

B]

0

5

10

15

20

0 5 10 15 20

Am

plitu

de [a

. u.]

0 1 2 3

t t1 2

Spe

ctru

m [d

B]

0

5

10

15

20

0 5 10 15 20

ff1 2

6 dB

Am

plitu

de [a

. u.]

0 1 2 3

Spe

ctru

m [d

B]

0

5

10

15

20

0 5 10 15 20 Time [µs] Frequency [MHz] Half-Power Bandwidth (-6 dB in pulse-echo mode) B f f= −2 1

Center Frequency f f fc = +12 2 1( )

Half-Power Pulse Length (50% in pulse-echo mode) τ = − ≈t t B2 1 1 /

Axial Resolution δ τ= 12 c

Radiation Pattern

Circular Piston Radiator

broadband (single cycle) narrow-band (five cycles)

Nor

mal

ized

Rad

ius,

r/

a

-2

-1

0

1

0 1 2 3 4

2

-10dB contour

θ-10 dB

Normalized Distance, z/N

near-field far-field

Na=

2

λ

Directivity Pattern Far-Field Radiation:

p r pe

rD kr

i k r( , ) ( , )θ θ= 0

Circular Piston Radiator in Fluid (Frequency-Dependent)

0o15o

45o30o

60o

90o

75o

90o

75o

60o

45o

30o15o

0o15o

45o30o

60o

90o

75o

90o

75o

60o

45o

30o15o

a / = λ 1.5

a / = λ 0.6

Piezoelectricity

Quartz (silicon dioxide, SiO2)

+

-+

-

-

+

- -

+

+

- +-

- -

+

+

+

+ + + + + + +

- - - - - - -+ + + + + + +

- - - - - - -

SiSi

Si

O OO

E bV

Fσ = A

Coupled Constitutive Equations:

S

E

eD E

e K S

ε = −σ

Typical Transducer Design

connector

housing

backing

piezoelectricdisk

matching layer &wear plate

electrodes

electrical lead

electricalnetwork

Piezoelectric materials:

Material Hr k Z Zw/ Q

Quartz (SiO2) 4.5 0.1 10.5 106 Lead Zirconate Titanate, PZT (Pb(Zr,Ti)O3)* 2,000 0.7 20 500

Barium Titanate (BaTiO3)* 1,200 0.5 20 500 Polyvinylidene Flouride (PVF2)* 12 0.14 2.7 25

*ferroelectric

Main Types of Piezoelectric Transducers

• immersion • contact • angle-beam • array • air-borne Specifics: • coupling (boundary) • matching (impedance) • damping (backing) • steering (rotation) • focusing (geometric)

Immersion Transducers

transducer

immersion tank

specimen

water

compressionalwave

shear or longitudinalwave

• coupling • matching ? • damping • steering • focusing

Impulse and Transfer Functions

f B P Pc a≈ ≈ = ≈5 116 23 550MHz MHz, . ( %), / %η

Time [1 µs/div]

Am

plitu

de [a

. u.]

Frequency [MHz]

Inse

rtio

n Lo

ss [d

B]

-40

-30

-20

-10

0

0 2 4 6 8 10

f B P Pc a≈ ≈ = ≈9 2 9 98 120. , ( %), / %MHz MHz η

Time [1 µs/div]

Am

plitu

de [a

. u.]

Frequency [MHz]

Inse

rtio

n Lo

ss [d

B]

-40

-30

-20

-10

0

0 5 10 15 20

Contact Transducers

transducer

specimen

couplant

Ref

lect

ion

Coe

ffici

ent

Frequency x Thickness [MHz mm]

0

0.2

0.4

0.6

0.8

1

air gap

-1010 -810 -610 -410 -210 010

water-filledgap

steel

R d≈ π ξ λ/ 0, where ξ = −Z Z Z Z0 1 1 0/ / • coupling ? • matching • damping • steering ? • focusing Ø

Angle-Beam Transducers

transducer

specimen

couplant

θs

θiwedge

c

cs

i

s

i= sin

sin

θθ

Plexiglas/Aluminum, longitudinal-to-shear transmission

Angle of Refraction [deg]

En

ergy

Tra

nsm

issi

o

00.10.20.30.40.50.60.7

30 40 50 60 70 80 90

"slip" boundary

"rigid" boundary

• coupling ? • matching • damping • steering ? • focusing ?

Air-Borne Piezoelectric Transducers

EV E

EV E

V = 0

• coupling • matching ??? • damping ? • steering • focusing ?

Electromagnetics Lorentz Force

G G G GF Q(E v B)= + ×

B

v

QFB

Ampère's law ∇ × = +G G

G

H JD

t

∂∂

Faraday's law ∇ × = −G

G

EB

t

∂∂

Ohm's law

G GJ E= σ

Je

conducting medium

Hp

He

Ip

Electro - Mechanic Conversion Transmission (I F→ ):

JeF

Bo

I

Reception (v V→ ):

v

Bo

V

Jc

Sensitivity and Polarization Lorentz Force

G G GF Q v BB o= ×

High Conductivity n I J dA Qve≈ =z Surface Traction τ = n I Bo Tangential Polarization:

Bon I

τ

Normal Polarization:

Bo n I

τ

EMAT Configuration I

spiral coil for radially polarized shear waves propagating normal to the surface

Bo

S

N

EMAT Configuration II

rectangular coil for linearly polarized shear waves propagating normal to the surface

Bo

SN

EMAT Configuration III

symmetric coil for longitudinal waves propagating normal to the surface

BoN S

Laser-Ultrasonics

with Specimen

Pulsed Laser

Interferometer

Fatigue Machine

Computer

Advantages: no mechanical contact 9 no need for couplant 9 absolute measurement 9 small detection aperture 9 broad bandwidth ? rough surface ? awkward shape ? moving object ? Disadvantages: expensive 9 low acoustic sensitivity ? mechanical instability ? low optical sensitivity ? surface damage ?

Heterodyne Laser Interferometer

∆ω =

Reference Mirror

Bragg Cell

Object

Detector

ΩB

Beam Splitter

Laser

ω

)

φ πλ

φo om= + 4oa tsin( )Ω

φ

P E E E E td o r o r B r= + + + −( ) cos(12

12

1 12 Ω oφ φ

E eoi tEo

φo= +1

( )

E eEr ri Ω tB r= − +

1( )ωt

λ optical wavelength ao vibration amplitude Ω acoustic angular frequency

Fabry-Perot Interferometer

Detector

ν0

Laser

Resonator

Object

νr

(t)v

Doppler shift:

ν ν θr tt

c( ) [ ]= +0 1 2cos

v( ), where ν λ0

86 10 500≈ × =MHz nm( )

a c≈ ≈ ≈ −1 0 3 10 9nm, at 5 MHz m / s,v v. /

Optical Frequency [100 MHz/div]

Tra

nsm

issi

o

= 98%= 0.5 m

RL

Optical Frequency [1 MHz/div]

Tra

nsm

issi

o

tuning

ν0

Laser Generation

Wavelength [µm]

Abs

orpt

ion

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10

aluminum

titanium

Low-Intensity Thermoelastic region: High-Intensity Ablation Region:

thermalexpansion plasma

laser beam

< 106 2W / cm

laser beam

> 106 2W / cm

recoil force

mostly tangential stress mostly normal stress

Linear Array Transducers

multiplexer

piezoelectric array

(amplitude & phase modulator) • axial scanning • no steering • apodization • geometrical lateral focusing • dynamic or static electronic axial focusing

Phased Array Transducers

amplitude & phase modulator

piezoelectric array

• sector scanning • electronic steering in two dimensions • apodization • dynamic or static electronic focusing

Ultrasonic NDE

Ultrasonics (high-frequency wave propagation in

idealized elastic media)

Wave-Material Interaction (special physical phenomena due to

interaction with imperfections)

Ultrasonic NDE

defect-free reflection, diffraction attenuation, velocity change

scattering, nonlinearity

defects cracks, voids

misbonds, delaminations isotropic anisotropy (orientation)

birefringence (polarization) quasi-modes (three waves) phase and group directions

residual stress effect

anisotropy texture

columnar grains prior-austenite grains

composites homogeneous incoherent scattering noise

attenuation dispersion (weak)

inhomogeneneity polycrystalline

two-phase porous

composite linear harmonic generation

acousto-elasticity crack-closure

nonlinearity intrinsic (plastics) damage (fatigue)

attenuation-free absorption viscosity, relaxation

heat conduction, scattering

elastic inhomogeneity geometrical irregularity

attenuation air, water, viscous couplants

polymers coarse grains

porosity

dispersion-free relaxation resonance

wave and group velocity pulse distortion

dispersion intrinsic (polymers)

geometrical (wave guides)

temperature-independent velocity change thermal expansion

temperature-dependence nonlinearity

residual stress (composites) phase transformation (metals) moisture content (polymers)

ideal boundaries flat, smooth,

rigidly bonded interface

mode conversion refraction, diffraction

scattering

imperfect boundaries curved, rough

slip, kissing, partial, interphase

canonical wave types plane wave

spherical waves harmonic

beam spread diffraction loss

edge waves spectral distortion

complex wave types apodization (amplitude)

focusing (phase) impulse, tone-burst

Inspection Principles and Techniques • Longitudinal, Shear, Rayleigh, Lamb, etc., Wave Inspection • Positive (backscattering) versus Negative (attenuation) Inspection • Pulse-Echo versus Pitch-Catch Inspection (through-transmission) • Contact versus Immersion Inspection • Normal Beam versus Angle Beam Inspection

transducer

specimen

couplant

transducer

specimen

couplant

θs

θiwedge

transducer

immersion tank

specimen

water

transducer

immersion tank

specimen

water

Pitch-Catch Inspection

transmitter receiver

specimen

receivertransmitter

specimen

receivertransmitter

immersion tank

specimen

water

Ultrasonic Flaw Detection

(Longitudinal, Positive, Pulse-Echo, Contact, Normal Beam)

Transmitter & Receiver

Ultrasonic Transducer

TestpieceReflected

Wave WaveIncident

EchoExcitation

FlawSignal

Advantages: high sensitivity high directivity depth ranging reproducible

Disadvantages: sensitive to geometry scanning requirement coupling is difficult

closed cracks can remain hidden

Basic Signal Processing

Time

Am

plitu

derf signal

Time

Am

plitu

de

rectified signal

Time

Am

plitu

de

weak smoothening

Time

Am

plitu

de

strong smoothening

Pulse-Echo Inspection

specimen

L

transmitter/receiver

d

t L cbw = 2 / t d cf = 2 /

Time [10 µs/div]

Am

plitu

de [1

0 dB

/div

]

backwall signals

"main bang"

tbw

tf

flaw signals

Pitch-Catch Inspection

specimen

L

transmitter

d

receiver

t L cbw = 2 /

Time [10 µs/div]

Am

plitu

de [1

0 dB

/div

]

backwall signals

"main bang"

tbw

Signal-to-Noise Ratio

signalnoise

coherent(material)

noise

incoherent(electrical)

Physical System

Time [2 µs/div]

Ultr

ason

ic S

igna

l [a.

u.]

FatigueCrack

noisy signal

averaging

averaging

synchronous

Grain Noise

texture-free (cast or annealed) material:

equi-axed grains, no preferred orientation

textured (forged, rolled, pressed, or drawn) material:

elongated grains, preferred orientation

cubic materials do not exhibit crystallographic texture

Grain Scattering Induced Attenuation in Polycrystalline Steel

(100 µm grain diameter)

logFrequency [MHz]

log

Atte

nuat

ion

Coe

ffici

ent [

dB/c

m]

-5

-4

-3

-2

-1

0

1

2

3

-1 0 1 2 3

Rayleigh region stochastic region geometrical region

shear

longitudinal

Measured Grain Scattering Induced Attenuation in SAE 1020 Steel

(longitudinal wave)

Frequency [MHz]

Atte

nuat

ion

Coe

ffici

ent [

dB/c

m]

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20

57 µm

48 µm

38 µm

31 µm

18 µm

10 µm

Surface Wave Flaw Detection

intermittent surface-breaking fatigue crack

length ≈0.035", depth ≈0.010"

Ti-6Al-4V specimen, 10 MHz

Time [1 µs/div]

Am

plitu

de [1

00 m

V/d

iv] smooth

rough

Ultrasonic Flaw Detection Below a Rough Surface

Rough Surface

Transducer

Flaw

Frequency [MHz]

Atte

nuat

ion

[dB

]

0

5

10

15

20

25

30

35

40

0 5 10 15 20

45.6 µm

25.6 µm

15 .2 µm

12.8 µm

11.4 µm

9.9 µm

8.7 µm

5.6 µm

Flaw Signals versus Artifacts

specimen

transducertransducerflaws

acoustic wave

Ultrasonic Probe

Cracked Rivet Hole

wave

Inspect at different orientations!

Nonlinearity

I material (stress-strain relationship)

Strain [a. u.]

Str

ess

[a. u

.]

Linear Limit

Elastic Limit

Ultimate Failure

II geometrical (strain-displacement relationship)

k k

F

Fε Fε

ε =+ −

≈a u a

a

u

a

2 2 2

22, F kε ε= , F

u Fa

ku

a= =2 3

3

Elastic Nonlinearity

Normalized Lattice Distance

Pot

entia

l Ene

rgy

[a. u

.]

0 1 2

typical

parabolic

potential well

Normalized Lattice Distance (Strain)

Ela

stic

Stif

fnes

s [a

. u.]

0.9 0.95 1 1.05 1.1

typical

parabolic potential function

unstrained

Acousto-Elasticity

c c( ) ( ...)σ η σ η σ= + + +0 1 22 ,

tension

d,pc

s,pc

d,nc

s,npc

s,nnc

σ

Five independent combinations of wave and polarization directions:

Wave Velocities in the Principal Directions

ρ λ µ σλ µ

λ λ µµ

λ µc md p, [ ( )]2 23 2

2 4 4 10= + ++

+ + + + +A

ρ λ µ σλ µ

λµ

λ µc md n, [ ( )]2 23 2

22

2= + ++

− + +A

ρ µ σλ µ

λµ

λ µc mn

s p, ( )2

3 2 44 4= +

++ + +

ρ µ σλ µ

λµ

λ µc mn

s np, ( )2

3 2 42= +

++ + +

ρ µ σλ µ

λ µµ

λc m ns nn, ( )2

3 2 22= +

+− + −

A, ,m nand Murnaghan coefficients

ρ density of the

σ tensile stress

material λ µ A m n

[109 Pa] [109 Pa] [109 Pa] [109 Pa] [109 Pa]

Aluminum 7064 59.3 27.4 -324 -397 -403

Armco iron 110 82 -348 -1030 1100

Polystyrene 2.9 1.4 -18.9 -13.3 -10

Pyrex 13.5 27.5 14 92 420

Longitudinal Velocity as a Function of Uniaxial Stress in 7064 Aluminum

Uniaxial Stress [MPa]

Long

itudi

nal V

eloc

ity [m

/s]

6360

6380

6400

6420

6440

6460

6480

-100 -50 0 50 100

parallel ( )

normal ( )

cd,p

cd,n

Excess Nonlinearity Due to Material Imperfections

η η η η η η ηtotal exc crack dislocG≈ + = + + +int int int

Crack Closure

Acoustoelastic Effect

x3

x1

x2

θ

n

β σ σ σ σ θ= − = + + + − −⊥ ⊥V V

VK K K K0

01 2 1 2

12

12

2( )( ) ( )( )cos|| ||

β σ σ( )0 1 2D = + ⊥K K||

β σ σ( )90 1 2D = +⊥K K||

β σY YK K= − + ⊥( )||

material Al 7064 Al 6061 Ni λ [109 Pa] 59.3 50.5 146 µ [109 Pa] 27.4 26 75

A [109 Pa] -324 -47.2 -673

m [109 Pa] -397 343 -757 n [109 Pa] -403 249 -168

σY [106 Pa] 368 256 460 K|| [10-12 Pa-1] -20.9 -22.0 +0.51

K⊥ [10-12 Pa-1] +9.7 +9.5 -9.1 ßY [%] +0.42 +0.32 +0.39

Ultrasonics vs Eddy Currents

7075 Aluminum, 5 MHz

Nor

mal

ized

Sur

face

Vel

ocity

2952

2956

2960

2964

0-5-10-15-20-25

0-40-80-120-160

External Stress [ksi]

External Stress [MPa]

1.000

1.001

1.002

1.003

Sur

face

Vel

ocity

[m/s

]

Al 7075, 200 kHz

Applied Stress [MPa]

Ele

ctric

al C

ondu

ctiv

ity [S

]

21.4

21.6

21.8

22

22.2

22.4

-600 -400 -200 0 200 400 600

Anisotropy

Cubic crystal structure

[100]

[010]

[001]

[110]

[111]

τxx

εxy= 0

τxx

εxy= 0/

Degree of Anisotropy

Anisotropy Factor

AC

C C=

−2 44

11 12 (unity for isotropic materials)

Ani

sotr

opy

Fac

tor

0

1

2

3

Sod

ium

Flu

orid

e

Yttr

ium

Iron

Gar

net

Fus

ed S

ilica

(Iso

trop

ic)

Tun

gste

n

Alu

min

um

Dia

mon

d

Sili

con

Iron

Nic

kel

Gol

d

Silv

er

longitudinal wave velocities in pure Nickel

[100] 5299 m/s

[110] 6027 m/s

[111] 6251 m/s

isotropic 6032 m/s

Velocity Distributions in the (001) plane

(1 km/s per divisions)

[100]

[001]

longitudinal

shear

Aluminum

[100]

[001]longitudinal

shear

Nickel

Longitudinal versus Shear

orientation versus polarization (birefringence)

°

Transducer

Specimen

dA

dB

Longitudinal Transducer

Specimen

d

pBShear

pA

"Fast" Mode "Slow" Mode

0°

22.5°

90°

45°

67.5°

"Fast" Mode "Slow" Mode

0°

90°

d°

Crystallographic vs Morphological Anisotropy

texture-free (cast or annealed) material:

equi-axed grains, no preferred orientation

textured (forged, rolled, pressed, or drawn) material:

elongated grains, preferred orientation

degree of texture: 1-10%

Earing During Deep-Drawing

Cold Pressing

Cold Drawing

soft axis stiff axis

"earing"

Hot (Cold) Pre-Processing

Texture Assessment by EMATs

Surface Wave Velocity Measurement

Textured Specimen

Transmitter Receiver

RayleighWave

Surface Wave Velocity Distribution

cold-pressed 2024 aluminum, 1.4 MHz, EMAT

2,850 m/s average velocity, 0.2% per division

0% (annealed) 0.45 %

0.8 % 1.6 %