ae03

Chapter 1. Introduction to AcousticEmission Testing . . . . . . . . . . . . . . 1

Part 1. Nondestructive Testing . . . . 2Part 2. Management of Acoustic

Emission Testing . . . . . . . . 13Part 3. History of Acoustic Emission

Testing . . . . . . . . . . . . . . . 21Part 4. Measurement Units for

Acoustic EmissionTesting . . . . . . . . . . . . . . . 25

Chapter 2. Fundamentals of AcousticEmission Testing . . . . . . . . . . . . . 31

Part 1. Introduction to AcousticEmission Technology . . . . 32

Part 2. Acoustic Emission Noise . . 41Part 3. Acoustic Emission Signal

Characterization . . . . . . . . 45Part 4. Acoustic Emission

Transducers and TheirCalibration . . . . . . . . . . . . 51

Part 5. Macroscopic Origins ofAcoustic Emission . . . . . . 61

Part 6. Microscopic Origins ofAcoustic Emission . . . . . . 69

Part 7. Wave Propagation . . . . . . . 79

Chapter 3. Modeling of AcousticEmission in Plates . . . . . . . . . . . 109

Part 1. Wave Propagation inPlates . . . . . . . . . . . . . . . . 110

Part 2. Formal AnalyticSolution . . . . . . . . . . . . . 114

Chapter 4. Acoustic Emission SourceLocation . . . . . . . . . . . . . . . . . . 121

Part 1. Fundamentals of AcousticEmission SourceLocation . . . . . . . . . . . . . 122

Part 2. Overdetermined SourceLocation . . . . . . . . . . . . . 131

Part 3. Waveform Based SourceLocation . . . . . . . . . . . . . 135

Chapter 5. Acoustic Emission SignalProcessing . . . . . . . . . . . . . . . . . 147

Part 1. Digital SignalProcessing . . . . . . . . . . . . 148

Part 2. Pattern Recognition andSignal Classification . . . . 157

Part 3. Classification of FailureMechanism Data fromFiberglass Epoxy TensileSpecimens . . . . . . . . . . . . 167

Part 4. Neural Network Predictionof Burst Pressure inGraphite Epoxy PressureVessels . . . . . . . . . . . . . . . 171

Chapter 6. Acoustic Leak Testing . . . 181Part 1. Principles of Sonic and

Ultrasonic LeakTesting . . . . . . . . . . . . . . 182

Part 2. Instrumentation forUltrasound LeakTesting . . . . . . . . . . . . . . 191

Part 3. Ultrasound Leak Testingof Pressurized Industrialand TransportationSystems . . . . . . . . . . . . . . 198

Part 4. Ultrasound Leak Testingof Evacuated Systems . . . 211

Part 5. Ultrasound Leak Testingof Engines, HydraulicSystems, Machinery andVehicles . . . . . . . . . . . . . . 213

Part 6. Electrical Inspection . . . . 215Part 7. Ultrasound Leak Testing

of Pressurized TelephoneCables . . . . . . . . . . . . . . . 218

Part 8. Acoustic EmissionMonitoring of Leakagefrom Vessels, Tanks andPipelines . . . . . . . . . . . . . 220

Acoustic Emission Testing

C O N T E N T S

Chapter 7. Acoustic Emission Testingfor Process and ConditionMonitoring . . . . . . . . . . . . . . . . 227

Part 1. Acoustic Emission Testingin Milling . . . . . . . . . . . . 228

Part 2. Acoustic Emission Testingof Resistance SpotWelding . . . . . . . . . . . . . 232

Part 3. Acoustic Emission WeldMonitoring of AluminumLithium Alloy . . . . . . . . . 235

Part 4. Acoustic Emission Testingfor Machinery ConditionMonitoring . . . . . . . . . . . 243

Part 5. Acoustic Emission Testingduring Grinding . . . . . . . 251

Part 6. Crack Detection duringStraightening of Axlesand Shafts . . . . . . . . . . . . 263

Chapter 8. Acoustic Emission Testingof Pressure Vessels, Pipes andTanks . . . . . . . . . . . . . . . . . . . . . 271

Part 1. Acoustic Emission Testingof Spheres and OtherPressure Vessels . . . . . . . . 272

Part 2. Acoustic EmissionTesting of CompositeOverwrapped PressureVessels . . . . . . . . . . . . . . . 281

Part 3. Acoustic Emission Testingof Pipelines . . . . . . . . . . . 284

Part 4. Acoustic Emission Testingof Delayed CokeDrums . . . . . . . . . . . . . . . 290

Part 5. Acoustic Emission Testingof Tank Floors . . . . . . . . . 296

Chapter 9. Acoustic Emission Testingof Infrastructure . . . . . . . . . . . . 305

Part 1. Acoustic Emission Testingof Bridges . . . . . . . . . . . . 306

Part 2. Acoustic EmissionMonitoring of CrackGrowth in Steel BridgeComponents . . . . . . . . . . 310

Part 3. Evaluation of SlopeStability by AcousticEmission Testing . . . . . . . 315

Chapter 10. Electric PowerApplications of AcousticEmission Testing . . . . . . . . . . . . 331

Part 1. Acoustic Emission Locationof Incipient Faults inPower Transformers . . . . 332

Part 2. Acoustic Emission Testingof High Energy SeamWelded Piping . . . . . . . . 342

Part 3. Acoustic EmissionMonitoring of LooseParts . . . . . . . . . . . . . . . . 347

Part 4. Acoustic Emission Testingof Steam Turbines . . . . . . 354

Chapter 11. Aerospace Applicationsof Acoustic Emission Testing . . 359

Part 1. Acoustic Emission Testingof Aircraft . . . . . . . . . . . . 360

Part 2. Fatigue Crack Monitoringof Aircraft EngineCowling in Flight . . . . . . 367

Part 3. Acoustic EmissionMonitoring of RocketMotor Case duringHydrostatic Testing . . . . 377

Part 4. Acoustic EmissionPrediction of Burst Pressurein Fiberglass EpoxyPressure Vessels . . . . . . . 383

Chapter 12. Special Applications ofAcoustic Emission Testing . . . . 391

Part 1. Acoustic Emission TestingUsing Moment TensorAnalysis . . . . . . . . . . . . . 392

Part 2. Acoustic Emission Testingfor Structural Design ofGrand Prix Cars . . . . . . . 401

Part 3. Acoustic EmissionMonitoring of Sand inPetroleum Wells . . . . . . . 408

Part 4. Active CorrosionDetection Using AcousticEmission . . . . . . . . . . . . . 415

Chapter 13. Acoustic EmissionTesting Glossary . . . . . . . . . . . . 427

Index . . . . . . . . . . . . . . . . . . . . . . . . . 435

Figure Sources . . . . . . . . . . . . . . . . . . . 446

Movie Sources . . . . . . . . . . . . . . . . . . . 447


Chapter 1. Introduction to AcousticEmission Testing

Movie. Discontinuities in steel . . . . 6Movie. Plastic deformation causes

cry of tin . . . . . . . . . . . . . . 21

Chapter 2. Fundamentals of AcousticEmission Testing

Movie. Acoustic emission differsfrom other methods . . . . . 32

Movie. Pencil break source . . . . . . 36Movie. Guard transducers control

noise . . . . . . . . . . . . . . . . . 42

Chapter 5. Acoustic Emission SignalProcessing

Movie. System with onechannel . . . . . . . . . . . . . . 150

Chapter 6. Acoustic Leak TestingMovie. Vibration at ultrasonic

frequencies of gas moleculesescaping from orifice . . . 183

Sound. Audible analog ofultrasonic signal . . . . . . . 194

Movie. Steam system leak test . . 202Movie. Amplitude rise heard

through ultrasounddetector as rough andraspy . . . . . . . . . . . . . . . . 214

Chapter 7. Acoustic Emission Testingfor Process and ConditionMonitoring

Movie. Discontinuities fromwelds . . . . . . . . . . . . . . . . 232

Chapter 13. Acoustic EmissionTesting Glossary

Movie. Pencil break source . . . . . 431


M U L T I M E D I A C O N T E N T S

Chapter 5. Acoustic EmissionSignal ProcessingFigure 1 Vallen-Systeme GmbH, Munich, Germany.

Chapter 7. Acoustic EmissionTesting for Process and ConditionMonitoringFigures 19-26 Holroyd Instruments, Bonsall Near

Matlock, Derbyshire, United Kingdom

Chapter 11. AerospaceApplications for Acoustic EmissionTestingFigures 1-11 ASTM International, West

Conshohocken, PA.

446 Acoustic Emission Testing

Figure Sources

Chapter 1. Introduction toAcoustic Emission TestingMovie. Discontinuities in steel Physical Acoustics

Corporation, Princeton, NJ; for the Federal HighwayAdministration, United States Department ofTransportation, Washington, DC.

Movie. Plastic deformation causes cry of tin PhysicalAcoustics Corporation, Princeton, NJ; for theFederal Highway Administration, United StatesDepartment of Transportation, Washington, DC.

Chapter 2. Fundamentals ofAcoustic Emission TestingMovie. Acoustic emission differs from other methods

Physical Acoustics Corporation, Princeton, NJ; forthe Federal Highway Administration, United StatesDepartment of Transportation, Washington, DC.

Movie. Pencil break source Physical AcousticsCorporation, Princeton, NJ.

Movie. Guard transducers control noise PhysicalAcoustics Corporation, Princeton, NJ; for theFederal Highway Administration, United StatesDepartment of Transportation, Washington, DC.

Chapter 5. Acoustic EmissionSignal Processing

Movie. System with one channel Physical AcousticsCorporation, Princeton, NJ; for the Federal HighwayAdministration, United States Department ofTransportation, Washington, DC.

Chapter 6. Acoustic Leak TestingMovie. Ultrasonic vibration of gas molecules escaping

orifice UE Systems, Elmsford, NY.Sound. Audible analog of ultrasonic signal UE Systems,

Elmsford, NY.Movie. Steam system leak test UE Systems, Elmsford,

NY.Movie. Amplitude rise heard through ultrasound detector

as rough and raspy UE Systems, Elmsford, NY.

Chapter 7. Acoustic EmissionTesting for Process andCondition Monitoring

Movie. Discontinuities from welds Physical AcousticsCorporation, Princeton, NJ; for the Federal HighwayAdministration, United States Department ofTransportation, Washington, DC.

Chapter 13. Acoustic EmissionTesting Glossary

Movie. Pencil break source Physical AcousticsCorporation, Princeton, NJ; for the Federal HighwayAdministration, United States Department ofTransportation, Washington, DC.

447Index

Movie Sources

Richard L. Weaver, University of Illinois, Urbana, Illinois

Modeling of AcousticEmission in Plates

3C H A P T E R

Background

Crack GrowthBrittle crack growth emits acoustic energyand acoustic emission testing of structuresis attractive for damage and remaining lifeassessment. Because it is often difficult todistinguish valid from false indications,conventional monitoring techniques areusually confined to representing eachevent by a small number of parameterssuch as energy, duration, frequency andtime of occurrence. For this reason, it isdifficult to distinguish events. Thetechniques used, in which a small numberof parameters are extracted from eachwaveform, neglect much of the detailedinformation presumably present in anoften very complicated waveform.

Technological developments inwaveform processing hardware andsoftware and in acoustic emissiontransducers have made it feasible to carryout fully quantitative acoustic emissionanalyses in real time. If the entirewaveform is analyzed in an elastodynamicpropagation model, the character andposition of the source may be extractablefrom that analysis. This approach wasproposed in the early 1980s butimplementation was expensive.1,2 Afterhardware and software improved, theproblem of theoretical elastodynamicanalysis of complex acoustic emissionwaveforms has been revisited.

Forward and Inverse ProblemsIn an arbitrary and complicated structure(consider for example a multiply rivetedand ribbed, curved shell), elastic wavepropagation is very complex. The forwardproblem of calculating a response from aspecified source in such a structure maybe treated in principle by finite elements.Such computations can be burdensome.Furthermore, such analysis provides noinsight into the important inverseproblem of identifying the source fromthe waveform. But if the structure is morehomogeneous (for example, a half spaceor a flat plate), other techniques can beused to solve the forward problem.

Even in a simple homogeneousstructure like a plate, elastic wavepropagation is complicated. There are two

distinct approaches commonly employedin plates, that of generalized rays and thatof guided modes. In a ray approach, eachreflection from a top or bottom surfaceand the associated generation of modeconverted rays must be considered. Thereare a large number of rays to track, evenafter a moderate propagation time. Theray picture is most useful only at shorttimes and distances. By implication, it isalso most relevant for frequencies thatexceed thickness resonances.

Calculation of elastic wave responses inisotropic uniform plates by means ofgeneralized rays has been treated3 and isnow widely carried out using a publishedalgorithm and software code.4 Thecomputations can become burdensomebecause of the need to track so manygeneralized rays at source-to-receiverdistances of several plate thicknesses.Computation time increases exponentiallywith the time to be studied. Nevertheless,the technique has been used because itcan calculate responses at arbitrarily highfrequencies with little additional effort.The chief application has been tolaboratory characterization of materials, ofreceivers and of simulated acousticemission sources.5,6 The techniques arenot useful for the calculation of wavesfrom realistic acoustic emission events inthe field over the distances and timestypical for such applications.

Wave ModesExpansion of responses in terms of theguided modes of wave propagationprovides an alternative technique ofcalculation,1,2,7-9 with a computationalburden that grows approximatelyquadratically with source-to-receiverdistance. Unlike the ray approach, higherfrequencies demand extra computationalresources. Thus, the technique is wellsuited for calculation of wave propagationover the distances and times typical forpractical acoustic emission. Mostimportantly, the technique yields analyticexpressions that can inform the importantinverse problem.2

One study1 considered theaxisymmetric problem of a concentratednormal step force applied to the surface ofa 10 mm (0.4 in.) thick glass plate and itsnormal surface displacement response at adistance of four to forty plate thicknesses.The work considered only the first eight


PART 1. Wave Propagation in Plates

branches of wave propagation, thuslimiting the accuracy to a frequency rangebelow 800 kHz.

That analysis is extended here toarbitrarily buried and arbitrarily orientedstrain nuclei sources, corresponding tobrittle growth of small cracks. Orientablesources like cracks will give rise tononaxisymmetric waves. In principle, theproblem posed here is much harder thanthe original problem that considered theaxisymmetric problem of a normallyoriented point surface force.1,2 However,because interest is restricted to responseswritten in terms of normal surfacedisplacements, the nonaxisymmetric partremains irrelevant. This is formally shownbelow, where the stated problem of thesurface normal displacement response to aburied strain nucleus source istransformed using reciprocity into theentirely equivalent problem of thecalculation of the strain response at aburied location to a step force on theplate surface.

Below, the mathematical problem isformally posed as a partial differentialequation with initial conditions andboundary conditions for a vector field uthat represents the elastodynamicdisplacement response to a buried strainnucleus source. The Greens dyadic for theplate is defined as the displacementresponse (at an arbitrary position) to anarbitrary, buried, concentrated step force.The posed problem is then solvedformally and exactly in terms of thatGreens dyadic. It transpires that it sufficesto consider the Greens dyadic for thenormal surface step force.

The Greens dyadic is then expressed interms of the normal modes of the plate,resulting in an expression for the desiredresponse as a sum over branches b, eachof which involves an integral with respectto wavenumber k. The integrand is givenas a closed form transcendental functionof the symmetry or antisymmetry of thebranch b; the wavenumber k; the distancer from source to receiver; the depth of theburied strain nucleus source; the time, thestrength and the orientation of thatsource and the implicit (and tabulated)solution of the dispersion relation b(k).

Problem FormulationIt is desired to calculate the linearelastodynamic response to a step strainnucleus in an infinite plate with stress freesurfaces.10,11 The notion of a strainnucleus is equivalent to a set of doubleforces, that is, double couples withoutmoment. Such nuclei are expected todescribe any pointlike source smaller thana wavelength. Of interest is the vertical(outward) normal displacement response

on one surface of the plate. The positionof the receiver is taken to be at thecoordinate origin (r = 0 in polarcoordinates) on the upper (z = +h) side ofthe plate of thickness 2h. The coordinatesystem is shown in Fig. 1. The source S islocated at position xS, with polarcoordinates rS = |xS|, = 0, z = zS. Thedynamic force distribution f (x,t) is givenas a strain nucleus12 (or moment tensor13)by the following function of x:

(1)

where the summation convention overrepeated indices is in force, where thesymmetric matrix M contains informationabout the orientation and strength of thesource and where the time dependencehas been taken to be (t), the unit stepfunction. Other time dependencies maybe obtained by suitable convolution withthe calculated responses. The forcedistribution may also be written as:

(2)

The sign convention used for ourdefinition of M is such that positiveprincipal values, or eigenvalues, of thematrix correspond to compressive loads.12

By way of illustration, it may be notedthat a suddenly broken bond that wasinitially under tension releases what hadbeen a compressive load on the

f t Mx

ti ill

x x x, = S

S( ) ( ) ( )3

f t Mx

ti ill

x x x, S( ) = ( ) ( )3

111Modeling of Acoustic Emission in Plates

FIGURE 1. Geometry of a plate with thickness 2h. A receiver,sensitive to vertical displacement, is placed on the topsurface above the coordinate origin. A transient stepfunction source acts at position rS, S, zS and is characterizedby its moment tensor M. Without loss of generality, theX axis is taken coincident with the line rs. Thus, S = 0.

Legendh = half plate thicknessMij = moment tensorrS = radial positionzS = depth positionS = axial position

Z

Y

X

Receiver rSMij zS

h

h

surrounding material. The force isequivalent to a constant compressive loadplus a suddenly applied tensile load.Because only the dynamic part is ofconcern, this is equivalent to only thesuddenly applied tensile part. Thus, sucha process is represented by a negativeprincipal value in M.

The following initial value boundaryvalue problem consists of the partialdifferential equation:

(3)

supplemented by quiescent initialconditions:

(4)

for all x and for t = 0. The partialdifferential equation is also supplementedby traction free boundary conditions:

(5)

at

where z = h for all r and . Of particularinterest is the displacement response v(t)at the position of the receiver, located onthe Z axis above the coordinate origin atthe top surface, where r = 0 and z = +h:

(6)

The tensor differential operator ij isdefined by:

(7)

where m indicates (xm)1. In thepresent case, the material density andthe material stiffnesses cijkl are presumedisotropic and homogeneous:

(8)

where and are the lam moduli.The desired response is the solution to

Eq. 3. It may be written in terms of theGreens dyadic Gjk for the plate, which isthe displacement response to a stepconcentrated force load. It is the solutionGjk to:

(9)

Gjk (x, xS, t) is the displacement responsein the j direction at position x to a unitstep force in the k direction at position xS.By taking a spatial derivative ( xSl)1 ofEq. 9 and multiplying by the matrix Mkl,the above becomes:

(10)

Equation 10 repeats the force distributionof a strain nucleus as described in Eq. 2.Thus, Eq. 10 indicates that the desiredwaveform, the response to a strainnucleus, is given in terms of G:

(11)

at r = 0 and z = +h. By reciprocity, spatialarguments and indices in G may beexchanged so the above is also equal to:

(12)

at r = 0, z = +h. Equation 12 indicates thatthe desired waveform is equal to M timesthe strain response at xS to a vertical stepload applied at r = 0 and z = +h.

This reciprocal form (Eq. 12) for theresponse (Eq. 11) will be especially useful,because it permits the calculation of thestrain response to a vertical step load atthe origin and equates that to the verticaldisplacement response to a strain nucleussource. Because vertical step loads areaxisymmetric, there is no component tothe displacement response and no dependence to that field. Thus, the strainsresulting from a vertical step force areconfined to the quantities rr, zz, rz andzr. (The strain plays a minor role.) Theoriginally stated problem may thereforebe recast more simply to calculate onlythe axisymmetric displacement responseu(x,t) to a step load on the upper surfaceat the origin1 and then to construct thelinear combination of its spatialderivatives:

v M Gt

kl kl

= S

S

( )3x x

x

, ,

v M G tkll

k= , ,S

S ( )x x x3

f t M Gt

M t

i ij kl jkl

ill

xx x

x

x xx

, , ,

( ) = ( )

= ( )

S

S

S

S

3

S Sij jk ikG t tx x x x, , ( ) = ( ) ( ) 3

=2

ij

ik ij kl il jk

k l

t

+ +( )

2

=ij ij k ikjl lt

c x x( ) ( )2

2

v t u r h t( ) ( )= = 0, z = +3 ,

z h=

3 0m =

u tu t

tjjxx

,,( ) ( )= = 0

= ,ij j iu t f tx x,( ) ( )


(13)

where the dummy variable xS is replacedwith x. The colon (:) represents acomplete inner product between thetensors.

The result is, by reciprocity, equal tothe originally stated problem concerningthe vertical displacement response at theorigin to a strain nucleus source M at x. Inthis restatement, it may be seen that thenonaxisymmetric modes do not enter nordo the shear horizontal waves. If thein-plane displacements caused by a strainnucleus source were required, it would benecessary to consider these waves too.

v M

Mu

Muz

Mu u

r

Mur

rrr

rzz

z

rzr

z

z

r

= :

=

+

+

+


The formal development of the responseof an infinite plate to a step vertical load(that is, the quantities u in Eq. 13) hasbeen described in the literature.1,2 Thedescription involved an expansion interms of the discrete normal modes of afinite plate of large radius R. If the(irrelevant) boundary conditions at R arechosen conveniently, then each of thesemodes belongs to a definite branch of therayleigh lamb dispersion relation (Fig. 2).The properties of such modes arediscussed in standard references.14-16 Therelevant formulas are summarized below.The plates eigen modes antisymmetricand symmetric with respect to z aredescribed separately.

Antisymmetric ModesAt wavenumber k and frequency , modesthat are antisymmetric with respect to z

and axisymmetric with respect to aregiven by Eqs. 14 to 23:

(14)

(15)

(16)

where J is a bessel function and prime ()indicates a derivative with respect toargument. Equations 14 and 15 define thefunction U(z) and W(z) for antisymmetricmodes. The coefficients B and C are givenby:

(17)

(18)

The antisymmetric rayleigh lambdispersion relation is:

(19)

The vertical wavenumbers and aregiven by:

(20)

(21)

where cd and ce are respectively thedilatational and shear (or equivoluminal)

22

22

=

ck

e

2

2

22

=

ck

d

0

1

4

2 2 2

2

= ( )= ( )+

R k

k h h

k h h

a

,

sin cos

cos sin

C k h= ( )2 2 sin B k h= 2 2 sin

u = 0

u W z J kr

B z Ck z

J kr

z = ( ) ( )= +[ ] ( )

a0

0

cos cos2

u U z J kr

B z C z

k J kr

B z C z

k J kr

r = ( ) ( )= [ ] ( )= [ ] ( )

a0

0

sin sin

sin sin

1


PART 2. Formal Analytic Solution

FIGURE 2. Dispersion curve for rayleigh lamb waves in a steelplate. The three straight dashed lines correspond to bulklongitudinal waves (the steepest line), bulk shear waves (theline of intermediate slope) and compressional waves in water(the line of least slope). The bold and unbold curves arerespectively the symmetric and antisymmetric branches.

10

9

8

7

6

5

4

3

2

1

0

0 2 4 6 8 10 12 14 16 18 20

Freq

uenc

y

(rat

io o

f di

latio

nal w

ave

spee

d to

thi

ckne

ss)

Wave number k = t 1, where t is thickness (arbitrary unit)

wave speeds of the plate material. There isa potential ambiguity in that the verticalwavenumbers and are defined only upto an unspecified sign. This ambiguity isof no consequence: the sign is arbitraryand final results are independent of thechoice of sign as long as it is madeconsistently. It may also be noted that and may be imaginary. This happens,respectively, whenever k > cd1 ork > ce1. In these cases, the usualidentities apply: sin i = i sinh ;cos i = cosh . In these cases, it mayalso happen that the modes themselvesbecome complex.

Equations 19 and 20 implicitlydetermine as a function of k. There is adiscrete infinity of solutions for eachreal k. If desired, they may be enumeratedby a branch index b = 1, 2 . . . .

As explained in previous work,1 if k isrestricted to a discrete set of values withsmall uniform spacing R1 chosen by therelation J0(kR) = 0 and if is one of thesolutions of the dispersion relation(Eq. 19), then these are vibration modesof the large radius (R ) disk withrigid-smooth boundary conditions at theouter rim. The mode has a squared norm:

(22)

where the integral is over the entirevolume of the disk shaped plate and Ma isgiven by an integral over the thickness ofthe plate, z going from h to +h:

(23)

Symmetric ModesSymmetric (with respect to z) butaxisymmetric (with respect to ) modeswith wavenumber k and frequency aregiven by:

(24)

(25)

The coefficients A and D are given by:

(26)

(27)

The symmetric rayleigh lamb dispersionrelation is:

(28)

As discussed elsewhere,1 these are the(symmetric and axisymmetric) modes ofthe large radius (R ) disk withrigid-to-smooth boundary conditions atthe outer rim, if k is restricted to a discreteset of values with uniform spacing R1and if is the one of the solutions of thedispersion relation (Eq. 28). This modehas a squared norm:

(29)

where the integral is over the entirevolume of the disk shaped plate andwhere Ms is:

(30) Ms s s

d

e

= ( ) + ( ) = { ( )

+ + ( )

+ }

U z W z dzA

h

ck

h

D kh

ck

h

ADk h h

2 2

22

22 2

2 22

22 2

2

22

22

4

sin

sin

cos sin

N

M

s

s

=

=

u udAdz

Rk

2

0

1

4

2 2

2

= ( )= ( )+

R k

k h h

k h h

s

,

cos sin

cos sin

D k h= ( )2 2 cos A k h= 2 2 cos

u W z J kr

A z Dk z

J kr

z = ( ) ( )= +[ ] ( )

s0

0

sin sin 2

u U z J kr

A z D z

k J kr

A z D z

k J kr

r = ( ) ( )= +[ ] ( )= +[ ] ( )

s0

0

cos cos

cos cos

1

Ma a

d

e

= ( ) + ( ) = + ( )

+

( )

+

U z W z dzB

h

Ck

h

C k

h

ck

h

BC k h h

a2 2

22

22 2

2 2

2

22 2

2

22

22

4

sin

sin

sin cos

N

M

a

a

=

=

u udA dz

Rk

2


The solutions of the dispersionrelations in Eq. 19 and Eq. 28 arediscussed in many places in theliterature.1,2,7,14-17 See also the equivalentformulas for anisotropic plates.18,19 Foreach k, there is a discrete infinity of realsolutions b(k) for each of the two cases ofsymmetric and antisymmetric modes.From each such solution b(k), it is simpleto obtain the values for the coefficients Band C or A and D and , and M.

Greens FunctionThe Greens dyadic (needed in Eq. 12 forthe desired waveform) can be expressed asa sum over the above normalizedvibration modes:

(31)

where the sum is over all of the eigenmodes m. The index m is equivalent, in amanner that need not be detailed, tospecification of the wavenumber k andthe branch b. The quantity N is thenormalization of the mode m;N = 2M Rk1.

Because the case j = 3 and sourceposition x are required at the pointr = 0, z = h, it may be noted thatnonaxisymmetric modes (none of whichwere specified above) do not contribute. Itis further noted that u3

(m)(x) = W(h) and(after Eq. 13):

(32)

where the sum is over all axisymmetricmodes: every branch b of the dispersionrelations above and every discrete value ofk along each branch. The sum may bereplaced with an integral along thebranch and a sum over branches:

(33)

So the response becomes:

(34)

where the integral over k runs from zeroto infinity and the sum is now a sum overthe infinite number of branches b of thedispersion relation, both symmetric andantisymmetric. The function (k) hasbeen given a subscript b to emphasize thatit is the solution of the dispersion relationcorresponding to the bth branch. The band k dependencies of W and U and Mhave been suppressed; similarly, thesuperscripts (a) and (s) on U and W aresuppressed.

The case of a step upward normalsurface force (equivalent for dynamicpurposes to a step release of a downwardnormal force) of magnitude F0 is similarto the above expression for the responseto a strain nucleus:

(35)

An extra factor of 1 must be inserted toeach antisymmetric branch in Eq. 35when the source and receiver are onopposite sides.

Equation 34 is the main result of thiscalculation. It permits numericalcalculations of the response of a normalsurface displacement detector to a distantburied oriented strain nucleus of stepfunction time dependence. In practice,the detector will not be ideal, so theabove time dependent v(t) must beconvolved with the transducer function.Similarly, it may be convolved with thetime derivative of the source function ifthe source function is thought to differfrom (t).

In practice, transducers may also benot pointlike. Under the assumption thatthe transducer does not affect thestructure and that the transducer isaxisymmetric around its own nominal

v k dk F W h

k t

k

b

b

b

= ( )

( )[ ]( )

cos

1

1

2

02

2

M

v k dk W h

M U z J kr

M W z J kr

M W z J kr

U z J kr

MU z

rJ kr

k t

b

rr r

zz z

rz r

z o

b

b

= ( )[ ] ( ) ( ){+ ( ) ( )+ ( ) ( )[+ ( ) ( )]+

( ) ( )

( )[ ]

cos

1

1

2

0

0

0

0

kk( )2M

m b

Rdk

v M U z J kr

M W z J kr

MU z

rJ kr M

W z J kr U z J kr

W ht

Rk

rr r

m

zz z

rz

r z

m

m

=

+

( ) ( ){ ( ) ( )

+( )

( ) + ( ) ( ) + ( ) ( )[ ]} ( )[ ] [ ]

00

0

0 0

1

2

cos

M

G t u u

t

ij i j

m

mm

m mx x x x, , =

( ) ( ) ( )

( )

( ) ( )1

2

cos

N


position, its spatial footprint may beaccounted for by inserting, into the aboveintegral, a factor of the hankel transformof the transducer footprint. In the case ofa piston receiver of radius R, this factor isessentially:

(36)

which has been normalized to unity for apoint sensor R 0.

The evaluation of Eqs. 34 and 35 hasbeen coded. The algorithm requires firstthe multibranched solution of therayleigh lamb dispersion relation on asufficiently fine grid in wavenumber kand out to a sufficiently large frequency. A separate code uses the instances of and k for each branch as obtained in thefirst code and then evaluates thequantities M, U(z), W(z) and theirderivatives, at the source coordinate z,and also W(h), for each branch b and eachwavenumber k along that branch by usingEqs. 14 to 30. The numerical integral isthen straightforward.

An example is shown (Fig. 3) for thepredicted out-of-plane displacement for acracklike acoustic emission source (set ofdouble sources) connected to the samesurface as the sensor. For this example,the source is 0.9 m (3 ft) from thetransducer.

Inverse ProblemEquation 34 may be rewritten in terms ofthe coefficients Q of distinct besselfunctions:

(37)

(38)

(39)

(40)

The response v(t) is then given by:

(41)

Thus, there are four distinct amplitudesneeded for each k and b. All four areneeded for an exact calculation ofwaveforms. They are functions of theparameters M that represent sourceorientation and strength; they arefunctions also of the depth of the sourcerepresented by the coordinate z and of thewavenumber and branch number of themode of interest. However, for an estimateof the relative importance of each modeat asymptotic distances rk >> 1, it sufficesto consider only a single effectiveamplitude Qeff for each type of strainnucleus source and each source depth andeach mode. At asymptotic distances,J0 (kr) is about J0(kr); J0 is /2 radians(90 deg) out of phase with J0 andJ0(kr)(kr)1 is negligible hence thedefinition of a single effective amplitude:

(42)

This single quantity represents thecontribution of a given mode {k,b}. It isuseful to plot it versus for eachbranch b, for each of several parametersM relating source orientation to strengthand for source coordinate z therebyillustrating the relative contributions ofdifferent sources to different modes. Ifsource character is to be ascertainedthrough examination of the waveform, itmust necessarily be by means of thisquantity Qeff; all information about thesource is transmitted through Qeff. Thisobservation forms the basis of the inverseprocedure below.

Q kk

k

Q k Q k Q k

b

eff ( ) = ( ) ( )[ ] + ( ) ( )[ ]

2

22

1 32

v k dk Q k J kr

Q k J kr Q k J kr

Q kJ kr

kr

k t

k

b

b

b

= ( ) ( )[+ ( ) ( ) + ( ) ( )+ ( ) ( )

( )[ ]( )

cos

1

1

1 0

2 0 3 0

40

2

Q k k M U zW h

4 2( ) ( ) ( )= M

Q k k M U zW h

rr3 2( ) ( ) ( )= M

Q k M kW z U z

W h

rz z2

2

( ) ( ) + ( )[ ]

( )=

M

Q k M zW h

zz z1 2( ) ( )[ ] ( )= M

2 22

1

RR

R

R

J kr r drJ k

k0 ( ) = ( )


FIGURE 3. Out-of-plane displacement prediction for acracklike source located 0.9 m (3 ft) from the transducer andconnected to the same surface, in Unified NumberingSystem G43400 alloy steel.

0.3

0.2

0.1

0

0.1

0.2

160 200 240 280 320 360 400 440 480 520

Dis

pla

cem

ent

(arb

itrar

y un

it)

Time (s)

The case of a step vertical surface load(Eq. 35) is also of interest:

(43)

where Q1stepload(k) is:

(44)

where sign = 1 if the branch is anantisymmetric branch and if the sourceand receiver are on opposite sides of theplate; sign = +1 otherwise. Thus,Q2 = Q3 = Q4 = 0 is identified for the stepforce source. The effective Q is given againby Eq. 42:

(45)

The inverse problem thus reduces toestimating, from measured waveforms,the effective modal amplitudes Qeff ateach {k,b}. A waveform in the timedomain at given time t may beunderstood as a superposition of themany modes {k,b} that propagate at thesame group velocity.2 At asymptoticallylarge source-to-receiver distances, the timeof signal arrival equals the time of sourceevent plus the ratio of distance to Vgb():

(46)

where Vgb is the group velocity:

(47)

Group velocity depends on plate materialand thickness and on the branch inquestion and the frequency of interest butnot on source type or distance. Time ofarrival is different for different branchesand frequencies.

Thus, at any time t, there are severalmodes contributing to the waveform, allwith different effective amplitudes Q andall with different frequencies. The result isa complex beat pattern. Extracting the

modal amplitudes Q from that pattern isdifficult because each such mode, havinga different frequency, has been affecteddifferently by the transducer sensitivity,the amplifier gain and the sourcefunction. Conversely, the waveform in thefrequency domain can be understood asan interference pattern between the manybranches that have the same frequencybut different amplitudes Q and differentarrival times. In either domain, thewaveform is complex, because there are somany interfering branches at the samepoint.

The best way to analyze thesewaveforms is in neither the frequencydomain nor the time domain but ratherby means of a simultaneous time andfrequency decomposition. If signalprocessing can effect this decomposition,the only remaining interferences would beat the occasional places where twobranches have the same group velocityand frequency.

Much signal processing literature triesto optimize such decompositions. Workhas focused on the spectrogram, that is,on the square of the short time fouriertransform, and on the wigner transform.The simplest analysis of time versusfrequency, the spectrogram, has beenfound to be adequate. A spectrogramcalculated from a theoretical waveform(an evaluation of Eq. 34) is shown inFig. 4, where amplitudes concentratedalong the loci of frequency versus arrivaltime are evident. Similar spectrograms are

Vdd k

b bg =

Time of signalarrival

Time of sourceevent

Distance

g

=

+ ( )Vb

Q kk Q k

kb

effstepload

( ) = ( )( )1

2

Q kW h

1

2

2stepload sign( ) = ( ) ( )M

vF

k dk

Q k J kr

k t

k

b

b

b

=

( ) ( )[ ]

( )[ ]( )

cos

0

stepload0

1

2

1


FIGURE 4. Spectrogram obtained from a theoretical acousticemission waveform. Amplitudes are confined to specific lociin the space of time versus frequency.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 100 200 300 400 500 600 700 800

Freq

uenc

y (M

Hz)

Time (s)

S0

A0

LegendS0 = asymmetric S waveA0 = symmetric P wave

shown in this NDT Handbook volumeschapter on source location. From suchspectrograms, the distance to the sourcecan be extracted by comparing the timesof arrival of different modes of knowngroup velocity. For example, it is clearthat the rayleigh wave arrives around timetR = 325 s. The S0 mode arrives aroundtS0 = 60 s. Thus, distance of the sourcecan be calculated:

(48)

By analyzing multiple points in suchplots, it is possible in principle to obtainhighly accurate source distances.

It is far more challenging to attempt tocharacterize the source. It has been foundthat source character can be determinedfrom the amplitudes in plots of timeversus frequency such as that of Fig. 4.The ability of such a plot to isolate asingle mode, and thereby estimate itseffective Q is critical. Comparison withtheoretical values of Qeff then permits adetermination of the source character. Theprocedure is described in detail in thisNDT Handbook volumes chapter onsource location.

Key to the technique is the recognitionthat time functions for source andreceiver are irrelevant if the effectiveamplitudes of different modes at the samefrequency are compared. If those modesarrive at different times, then they do notinterfere and a plot like that of Fig. 4permits the evaluation of the ratio ofeffective Qs. A set of 13 mode pairs hasbeen identified for which one frequencyarrives at two distinct times and no othermodes arrive at that time, with thatfrequency.

For example, the color version of thescreen shown in Fig. 4 displays theamplitude of the S0 and A0 modes at afrequency of 0.1 MHz. Under theseconditions, there is no interference andthe effective mode amplitudes can beestimated without ambiguity from thestrength of the corresponding points ofthe spectrogram. Although the amplitudeitself is a function of the source strengthand time function and receivercharacteristics, the ratio of effective Qs ofdifferent modes of the same frequency is afunction of only source character andsource depth. Those many ratios are thencompared to the values of those ratios foreach source in a catalog of candidatesources and the best fit candidates arepresented together with their fit quality.

Success with real waveforms, possiblywith limited bandwidth and perhapsshorter distances from source to receiver isnot documented in the present

discussion. Source orientation and depthwere recovered from waveforms asgenerated by Eq. 34 or 35 and from theirspectrograms like that of Fig. 4. Successwas best when distance from source toreceiver was greater than 100 times theplate thickness. This is perhapsunsurprising because large distances leadto better separation of modes withdifferent group velocity.

It is clear that the best resolution ofsource type will be obtained if waveformsare wideband enough to encompassseveral of the 13 mode pairs identified asmost useful. These mode pair frequenciesrange up to about the cutoff of the tenthbranch, that is, to a frequency of about400 kHz in 25 mm (1 in.) thick steel.

1 1 S

S

dt t

c c

R

R

0

0


1. Weaver, R.L. and Y.-H. Pao.Axisymmetric Waves Excited by aPoint Source in a Plate. Journal ofApplied Mechanics. Vol. 49.New York, NY: ASME International(1982): p 821-836.

2. Weaver, R.L. and Y.-H. Pao. Spectra ofTransient Waves in Elastic Plates.Journal of the Acoustical Society ofAmerica. Vol. 72. Melville, NY:American Institute of Physics, for theAcoustical Society of America (1982):p 1933-1941.

3. Ceranoglu, A.N. and Y.-H. Pao.Propagation of Elastic Pulses andAcoustic Emission in a Plate. Journalof Applied Mechanics. Vol. 48.New York, NY: ASME International(1981): p 125-147.

4. Hsu, N.N. Dynamic Greens Functions ofan Infinite Plate A Computer Program.NBSIR 85-3234. Gaithersburg, MD:National Institute of Standards andTechnology (1985).

5. Breckenridge, F., T. Proctor, N.N. Hsu,D. Fick and D. Eitzen. TransientSources for AE Work. Progress inAcoustic Emission V. Tokyo, Japan:Japanese Society for NondestructiveInspection (1990): p 10-37.

6. Sachse, W. Recent Developments inQuantitative AE. Proceedings of theIUTAM Symposium on Elastic Waves andUltrasonic Nondestructive Evaluation[1989]. Amsterdam, Netherlands:Elsevier, for the International Union ofTheoretical and Applied Mechanics(1990).

7. Santosa, F. and Y.-H. Pao. TransientAxially Asymmetric Response of anElastic Plate of Finite Thickness. WaveMotion. Vol. 11. Amsterdam,Netherlands: Elsevier (1999):p 271-295.

8. Mal, A.K. and T.C.T. Ting, eds. WavePropagation in Structural Composites.AMD-90. Materials Park, OH: ASMEInternational (1988).

9. Mal, A.K. Wave Propagation inLayered Composite Laminates underPeriodic Surface Loads. Wave Motion.Vol. 10. Amsterdam, Netherlands:Elsevier (1988): p 257-266.

10. Pao, Y.-H. Theory of AcousticEmission. Elastic Waves andNondestructive Testing of Materials.AMD-29. New York, NY: ASMEInternational (1978): p 107-128.

11. Pao, Y.-H. Elastic Waves in Solids.Journal of Applied Mechanics. Vol. 50.New York, NY: ASME International(1983): p 1152-1164.

12. Love, A.E.H. A Treatise on theMathematical Theory of Elasticity.New York, NY: Dover (1927).

13. Aki, K. and P. Richards. QuantitativeSeismology Theory and Methods. SanFrancisco, CA: W.H. Freeman (1980).

14. Achenbach, J.D. Wave Propagation inElastic Solids. Amsterdam, Netherlands:North Holland (1973).

15. Miklowitz, J. Theory of Elastic Wavesand Waveguides. Amsterdam,Netherlands: Elsevier/North Holland(1978).

16. Graff, K.F. Wave Motion in Elastic Solids.New York, NY: Dover (1975).

17. Weaver, R.L. Diffuse Waves in FinitePlates. Journal of Sound and Vibration.Vol. 94. San Diego, CA: AcademicPress (1984): p 319-335.

18. Nayfeh, A. and D. Chimenti.Ultrasonic Wave Reflection fromLiquid Coupled Orthotropic Plateswith Applications to FibrousComposites. Journal of AppliedMechanics. Vol. 55. New York, NY:ASME International (1988): p 863-870.

19. Banerjee, S., A.K. Mal andW.H. Prosser. Analysis of TransientLamb Waves Generated by DynamicSurface Sources in Thin CompositePlates. Journal of the Acoustical Societyof America. Vol. 115, No. 5. Melville,NY: American Institute of Physics, forthe Acoustical Society of America(2004): p 1905-1911.


References

Table of Contents with Chapter LinksMultimedia ContentsChapter 3. Modeling of Acoustic Emission in PlatesPart 1. Wave Propagation in PlatesBackgroundProblem FormulationFigure 1

Part 2. Formal Analytic SolutionAntisymmetric ModesSymmetric ModesGreen's FunctionInverse ProblemFigure 2Figure 3Figure 4

References

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