AEEM-7028

Ultrasonic NDE

Part 1

Introduction to Ultrasonics

What is ultrasonics?

Ultrasonics is a branch of acoustics dealing with the generation and use of (generally) inaudible acoustic waves.

low-intensity applications:

to convey information through a system

to obtain information from a system

1 3

avg 10 10 WP − −≈ −

2 4

peak 10 10 WP ≈ −

5 1010 10 Hzf ≈ −

high-intensity applications:

to permanently alter a system

2 4

avg 10 10 WP ≈ −

4 510 10 Hzf ≈ −

What is Nondestructive NDE?

Methods of NDE

Visual

Liquid Penetrant

Magnetic Particle

Eddy Current

Ultrasonic

X-ray

Microwave

Acoustic Emission

Thermography

Laser Interferometry

Replication

Flux Leakage

Acoustic Microscopy

Magnetic Measurements

Tap Testing

Terminology:

ultrasonic nondestructive inspection (NDI)

ultrasonic nondestructive testing (NDT)

ultrasonic nondestructive evaluation (NDE)

Main Fields of Ultrasonic NDE:

material production processes

material integrity following transport, storage and fabrication

the amount and rate of degradation during service

Ultrasonics versus 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

Simple Harmonic Wave

x

λ

t = t1t2

t3

cu

u0

-u0

0( , ) cos[ ( ) ]xu 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 ct= − Oscillatory wave packet

( )cos[ ( )]u f x ct k x ct= − −

Fundamental Wave Modes Bulk Waves: Longitudinal Wave:

wavedirection

Shear Wave:

wavedirection

Guided Waves:

e.g., Surface Wave:

wavedirection

rod waves, Lamb waves in plates, etc.

Static Uniaxial Load

L

A

PP

L + u

1. Deformation

planes originally normal to the axis remain normal, but their separation changes 2. Strain

uL

ε =

3. Stress

Eσ = ε , ε = − νε 4. Load

AP dA A= σ = σ∫

5. Displacement

P LuE A

=

Fundamental Longitudinal Mode

( )u u x=

dxx

u

dx

σ ∂σσ + dx∂x

Equation of motion:

2

2( ) udx A A Adxx t

∂σ ∂σ + −σ = ρ∂ ∂

2

2u

x t∂σ ∂= ρ∂ ∂

A cross-sectional area

ρ mass density

Constitutive equation:

Eσ = ε

ε axial strain

E Young's modulus

Displacement-strain relationship:

ux

∂ε =

∂

Wave equation:

2 2

2 2u uE

x t∂ ∂

= ρ∂ ∂

2 2

2 2 2rod

1u ux c t∂ ∂

=∂ ∂

longitudinal wave velocity in a thin rod

rodEc =ρ

rodsin[ ( )]u A k x c t= −

rod( )u f x c t= −

Solution of the General Wave Equation

2 2

2 2 21u u

x c t∂ ∂

=∂ ∂

where c is the wave velocity:

stiffnessvelocitydensity

=

Propagating harmonic wave represents a solution of the wave equation:

0( , ) cos[ ( ) ]xu x t u tc

= ω − + ϕ

Arbitrary wave pulse of the general form ( , ) ( )xu x t f tc

= − also satisfies the wave

equation:

2

2 ( , ) ''( )xu x t f tct

∂= −

∂

2

2 21( , ) ''( )xu x t f t

cx c∂

= −∂

Static Torsional Load

circular cross section (no warping):

LJ

θ

d

TTγ

1. Deformation planes originally normal to the axis remain normal, their separation remains the same, but

they rotate in their in their own plane, i.e., around to the axis 2. Strain

rLθ

γ =

3. Stress Gτ = γ , where 2(1 )E G= +ν

4. Load 2

A A

G G JT r dA r dAL Lθ θ

= τ = =∫ ∫

2

AJ r dA= ∫∫ polar moment

5. Displacement

T LG J

θ =

Static Torsional Load (cont.)

arbitrary cross section (warping)

TT

t

T LG J

θ =

tJ twisting moment

Saint-Venant approximation

4 4

2 404tA AJ

JJ≈ ≈

π

Fundamental Torsional Mode

fundamental mode: ( )xθ = θ

dx

θT + dTT

γ + dθ θ

Equation of motion: 2

2dT J dxt

∂ θ= ρ

∂

tT G Jx∂θ

=∂

2 2

2 2tG J Jx t∂ θ ∂ θ

= ρ∂ ∂

2 2

2 2 21

tx c t∂ θ ∂ θ

=∂ ∂

t tt s

G J Jc cJ J

= =ρ

Fundamental String Mode

no bending moment

Ax

vy,

x

T + dTζT

dx

dθvy,

Curvature:

dx d= ζ θ

1''ddxθ= =

ζv

Equation of motion: 2

2T d Adxt

∂θ = ρ

∂v

2 2

2 2T Ax t∂ ∂

= ρ∂ ∂v v

2 2

2 2 2string

1x c t∂ ∂

=∂ ∂v v

stringTc σ

=ρ

TTA

σ = tension stress

Simple Bending Deformation

Ix

vy,

xM M

ζvy,

1. Deformation

planes originally normal to the axis remain normal, on the average their separation remains the same, but they rotate around the axis of moment

2. Strain

''y yε = − = −ζ

v

3. Stress

Eσ = ε 4. Load

2'' ''A A

M y dA E y dA E I= − σ = =∫ ∫v v

' '''V M E I= − = − v

' ''''q V E I= − = v

5. Displacement

2''dx= ∫∫v v

Fundamental Flexural Mode in a Thin Rod

4

4I E qx

∂=

∂v

I moment of inertia

v transverse displacement

q distributed load intensity for a unit length

Inertia forces decelerating the beam

2

2q At

∂= − ρ

∂v

A cross-sectional area

For harmonic vibrations

sin[ ]k x t= − ωov v

4 4 4x k∂ ∂ =/

2 2 2t∂ ∂ = −ω/

4 2I E k A= ρ ω

Dispersion of Flexural Waves propagating modes:

24f

I Eck Aω ω

= = ±ρ

non-propagating modes:

24f

I Ec ik Aω ω

= = ±ρ

For a rectangular bar of height h

2 24 rod0.5373

12fE hc c hω

= = ωρ

rodc E= ρ/ longitudinal wave velocity in the thin rod

For a thin plate of thickness h

E → 2(1 )E − ν/

2 24

212(1 )fE hc ω

=− ν ρ

For a cylindrical rod of diameter d

2 16I A d=/ /

rod0.5fc c d= ω

Which Guided Wave Mode To Use?

• generation/detection

• loading/damping/leaking

• dispersion

• vibration profile/distribution

Wave Types in Solids and Fluids Extended (bulk) fluid medium:

longitudinal (compressional, dilatational, pressure) Extended (bulk) solid medium:

longitudinal (dilatational)

shear (transverse, equivoluminal) Solid half-space:

surface (Rayleigh) wave Solid plate:

plate (Lamb) waves

Rods, strings, etc.

stiffnessvelocitydensity

=

Dilatational Modes thin rod aligned with the x-direction ( 0y zσ = σ = )

x xEσ = ε

rodEc =ρ

thin plate parallel to the x-y plane ( 0y zε = σ = )

21x xE

σ = ε− ν

ν Poisson's ratio

rodplate rod2 2

1.05 (for 0.3)(1 ) 1

cEc c= = ≈ ν =− ν ρ − ν

infinite medium ( 0y zε = ε = )

2(1 )

(1 )(1 2 )211

x x xE E − ν

σ = ε = ε+ ν − νν−

− ν

rod rod(1 ) (1 ) 1.16

(1 )(1 2 ) (1 )(1 2 )dEc c c− ν − ν

= = ≈+ ν − ν ρ + ν − ν

Acoustic Waves in a Gas

p RT= ρ gas equation

T absolute temperature

R gas constant For an adiabatic process

p K γ= ρ

bulk modulus

B p K γ= ρ∂ ∂ρ = γ ρ/

d Bc = ρ/

od

pRTc

γ= = γ

ρ

po static (ambient) pressure

p vc cγ = / specific-heat ratio

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

x

σx

σy

σx-

σy-

y

ux

-τ

x

τyx

yx

τxy−τxy

uy

xy xyτ = μγ (µ = G)

y

xyux

∂γ =

∂

2 2

2 2 21y y

s

u u

x tc

∂ ∂=

∂ ∂

sc μ=

ρ

2 21 2

d

s

cc

− ν=

− ν, 2

1plate

s

cc

=−ν

, 2(1 )rod

s

cc

= +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 Aet

− ω∂= = − ω

∂v

( )x i kx tx xx xx

uC C Ai k ex

− 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

d dZ c= ρ

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. Similarly, for shear waves

s sZ c= ρ

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

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 2

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

u u t t t tω +ω ω −ω+ = ω + ω =

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

2cos( )cos( )2 2

u x t kx t k k x tkk x t x t

= − ω + + δ − ω + δωδ δω

≈ − ω −

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:

Phase Velocity versus Group Velocity Carrier or phase velocity

2

2

pc k kk

δωω + ω= ≈

δ+

Envelope or group velocity

gck k

δω ∂ω= →

δ ∂

Characteristic equation

( , ) 0pF c k = or ( )p pc c k=

pk cω =

p

g pc

c c kk

∂= +

∂

Dispersion equation

( )p pc c= ω

1

pg

p

p

cc c

c

=∂ω

−∂ω

Spectral Representation In the case of dispersive wave propagation,

( ) becomes ( )( )

x xf t f tc c

− −ω

Let us assume that f(t) is known at x=0. Fourier transform:

{ ( )} ( ) ( ) exp( )f t F dt f t i t∞

−∞= ω = − ω∫F

Inverse Fourier transform:

1{ ( )} ( ) ( ) exp( )2

F f t d F i t∞

−∞ω = = ω ω ω∫

πF -1

Shift theorem:

{ ( )} ( ) exp( )p pf t t F i t− = ω − ωF

Dispersive wave propagation:

( , ) ( ,0) exp[ ] ( ,0) exp[ ( )]( )xF x F i F i xk

cω = ω − ω = ω − ω

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