2-1 AE Parameters

2-2 Source Mechanisms - Elastodynamics

2-3 Source Mechanisms - Crack Modeling

2-1 AE Parameters2-1 AE Parameters 　　　　　　　　　　　　　　　　 October 11, 2007 (1/3)October 11, 2007 (1/3)

Analog and DigitalIdentification of AE SignalAE Signal ParametersParametric Analysis (1) Kaiser effect (2) Crack classification (3) Amplitude fistribution

IntroductionIntroduction

Final goal of monitoring AE phenomena is to provide beneficial information to prevent fatal fracture, by correlating detected AE signals with growing fracture process or deterioration.

AE activity is observed transiently or unexpectedly, and the signals generally contain higher frequency components over the audible range as well as a variety of durations.

Conversion from waves to Conversion from waves to electrical signalselectrical signals

Parametric analysis means “analog” signal processing. It is noted that original waves are three-dimensional, but

signals are just one-dimensional.

Basic treatmentBasic treatment A signal triggering is conventionally made by setting

threshold. In the case of trigger-monitoring, only the signals, of which amplitudes exceed the threshold levels, are recognized as AE signals.

In the early age of AE measurement, the performance of transient waveform-recorders was so poor as that parametric features of “analog” signals were normally employed for evaluating AE characteristics.

These are hit, amplitude, counts, duration and so forth.

Analysis updateAnalysis update

Currently, as rapid progress of computer technology, AE waveforms can be recorded readily as well as the parametric features.

Such waveform-based features as peak frequency and frequency centroid are additionally determined in real time from the fast Fourier transform (FFT) of recorded waveforms.

AE parametric features are extracted and provide good information to correlate the failure behavior of materials.

Identification of AE SignalIdentification of AE Signal Different from detected waves in

ultrasonic or vibration tests, AE signals emerge rapidly and randomly. Discrimination of AE signals from running waves is the first step for analyzing AE activity.

To this end, the voltage threshold of AE wave, which is equivalent to a voltage level on an electronic comparator, is set. The signals which exceed the voltage threshold are identified as AE signals.

One waveform signalOne waveform signal

Termination of the AE signal or the duration is determined as the period when the signal do not exceed the voltage threshold.

Conventionally, the duration is defined by users and set as a constant time (e.g. hit-lockout time or dead time)

In the case of digital recording, the start time to record the waveform is the same as the parametric feature extraction by means of the voltage threshold.

The length or duration of a waveform to be recorded is determined independently by users.

Digital treatmentDigital treatment Users shall determine

the both of sampling time and total number of samples in addition to the voltage threshold.

For example, the waveform length of 1024 microseconds is set up as 1 MHz sampling rate in 1024 samples.

t: sampling time 1/t: sampling frequency f = 1/(Nt) :frequency increment Fny=1/2t : Nyquist frequency

Frequency range optimal

up to Fny/3 – Fny/5

Signal ParametersSignal Parameters[ISO 12716 2001].[ISO 12716 2001].

Conventional parameters-1Conventional parameters-1 Hit: a signal that

exceeds the threshold and causes a system channel to accumulate data.

It is frequently used to show the AE activity with counted number for a period (rate) or accumulated numbers.

Count/ring-down count: the number of times within the duration, where one signal (waveform) exceeds a present threshold.

“Count” is also employed to quantify the AE activity as well as “hit”. It is noted that “counts” depend strongly on the employed threshold and the operating frequency.

Conventional parameters-2Conventional parameters-2 Amplitude: a peak

voltage of the signal waveform is usually assigned - Max. amp.

Amplitudes are expressed on a decibel scale instead of linear scale where 1mV at the sensor is defined as 0 dBAE.

The amplitude is closely related to the magnitude of source event.

the amplitude is also important parameter to determine the system’s detectability.

Generally the detected amplitude shall be understood as the value does not represent the emission-source but the sensor response after losing the energy due to propagation.

Conventional parameters-3Conventional parameters-3 Duration: a time interval

between the triggered time of one AE signal (waveform) and the time of disappearance is assigned.

The duration is expressed generally on microseconds, which depends on source magnitude and noise filtering.

Rise time: a time interval between the triggering time of AE signal and the time of the peak amplitude is assigned.

The rise time is closely related to the source-time function, and applied to classify the type of fracture or eliminate noise signals.

AE energy (RMS)AE energy (RMS)

Energy: definitions of energies are different in AE system suppliers, but it is generally defined as a measured area under the rectified signal envelope (Root-Mean-Square).

The energy is preferred to interpret the magnitude of source event over counts, because it is sensitive to the amplitude as well as the duration, and less dependent on the voltage threshold and operating frequencies.

Other parametersOther parameters Average frequency: a

calculated feature obtained from “Count” divided by “Duration”, which determines an average frequency over one AE hit.

Initial frequency: a calculated feature derived from “Count to Peak” divided by “Rise time”.

Reverberation frequency: a calculated feature derived from “Count-Count to Peak” divided by “Duration-Rise time”.

RA value: a calculated feature derived from ”Rise time” divided by “Amplitude”, showing the reciprocal of gradient in AE signal waveforms, which is reported in ms/V.

Frequency parametersFrequency parameters Frequency centroid: a sum of

magnitude times frequency divided by a sum of magnitude, as equivalent to the first moment of inertia.

Peak frequency: a frequency defined as the point in the power spectrum at which the peak magnitude is observed.

Parametric AnalysisParametric Analysis

In order to interpret acquired AE data, correlation-based, time-based or external parameter-based AE parametric features are customarily used with their occurrence rate or accumulated trend.

Statistical values of the parameter and some combinations among AE parameters as well as external parameters have been studied intensively for relating to the scale of fracture or the degree of damage in the materials/structures.

Kaiser effectKaiser effect Felicity ratio = PAE/P1st

where PAE is a stress at which AE activity starts to generate, and P1st is the maximum stress.

The Felicity ratio becomes equal to or larger than one in an intact or stable state, while in a damaged condition it reveals smaller than one.

Famous irreversibility effect of AE occurrence.

AE activity is seldom observed until the load exceeds the previous load level.

Based on the Kaiser effect, T. Fowler proposed the Felicity ratio, which can show the damage quantitatively in tank structures.

Results of loading test at a wharfResults of loading test at a wharf

Damage qualification by Damage qualification by load ratio and calm ratioload ratio and calm ratio

Load ratio = load at the onset of AE activity in the subsequent loading / the previous load.

Calm ratio = the number of cumulative AE activity during the unloading process/ total AE activity during the whole loading in each cycle.

Heavy damageIntermediate

damage

Intermediatedamage

Minor damage

Load ratioC

alm

rat

io

Crack classificationCrack classification

Classification of crack types is proposed, using the combination of the average frequency and the RA values.

This classification technique has been standardized [JCMS 2003]

Amplitude distributionAmplitude distribution

Gutenberg-Richter Relation on earthquakes

Log N = a – bM M=LogA: Magnitude

Log N = a - bLogA

b-value Large:small amp.

Small:large amp.

RemarksRemarks

In AE parametric analysis, parametric features are mostly derived by analog processing.

Based on definitions and properties of AE parameters, signal processing to evaluate the fracture processes is to be selected.

Results obtained by applying these parameters are delivered in applications.

2-2 Source Mechanisms – 2-2 Source Mechanisms – Elastodynamics Elastodynamics October 11, 2007 (2/3)October 11, 2007 (2/3)

Field EquationIntegral Representation Green’s Functions

Introduction of ElastodynamicsIntroduction of Elastodynamics

In order to discuss source mechanisms of AE, elastic waves due to a micro-crack nucleation in a homogeneous medium are theoretically studied.

Although many materials are not homogeneous

but heterogeneous, material properties in elastodynamics are fundamentally dependent on the characteristic dimensions of materials.

Basics - BackgroundBasics - Background It is reasonable to refer to concrete and rock as

homogeneous in AE measurement. This is because the dynamic heterogeneity is closely dependent on the relation between the wavelengths and the characteristic dimensions of the materials.

In the case that the wavelengths are even longer than the sizes of heterogeneous inclusions, the effect of heterogeneity is inconsequent.

- Basics of scattering theory -

ExampleExample In the case of AE waves in concrete or rock, the

velocities of elastic waves are over 1000 m/s. Thus, the use of frequency range up to some 100 kHz corresponds to the case where the wavelengths are longer than several centimeters. It results in the fact that concrete consisting of only normal aggregate (with around 10 mm diameter) or rock of minerals is reasonably referred to as homogeneous.

Wavelength = velocity of medium / frequency

= vp/f = 1000 [m/s] /100 x1000 [1/s] = 0.01 [m]

The case of thin specimenThe case of thin specimen It is noted that the wavelengths detected should be

physically observed or detected in the propagating medium. This is not the case of AE waves in a thin plate, because the thickness of the plate is occasionally shorter than the wavelengths.

If the plate is made of steel, the use of 1 MHz frequency range corresponds to detecting AE waves of around 5 milli-meter wavelengths. As a result, the wavelengths of propagating waves are often longer than the thickness.

Guided wavesGuided waves

Thus, diffracted and dispersive waves are generated as guided waves.

Because the amplitudes of these waves are dominantly larger than those of P (primary or longitudinal) and S (secondary or shear) waves, P and S waves are normally smeared or neglected in the AE detection of thin plate samples.

The dispersive waves are defined as waves of which velocities are dependent on frequency components (wavelengths).

AE theory based on plate wavesAE theory based on plate waves

Recent theories of AE in metal are based on the guided wave.- no relation to source motions.

One example is the Lamb wave (surface wave) observed in a plate. This is one key factor for the flaw location in the plate-like members.- Do not insist on the velocities of P wave or S wave, they are normally slower than them.

Integral RepresentationIntegral Representation To derive equations, Gauss's integral theorem is

definitely necessary.

  ∫Vf,j (x)dV = ∫S f(x) nj dS,

Basics of the tensor notation f,j = ∂f/∂xj (x1, x2, x3) instead of (x,y,z)

Vector a = aiei --- ai

Body waves (P and S) Body waves (P and S) Wave equation :

[+]uj,ij(y,t) + ui,jj(y,t) = ∂2ui(y,t)/∂t

(1) Consider the volumetric strain: ui,i

[+]uj,iji(y,t) + ui,jji(y,t) = ∂2ui,i(y,t)/∂t

[+2]uj,jii(y,t) = ∂2ui,i(y,t)/∂t

Velocity of P wave : vP = [+2/]1/2

(2) Consider without the volume strain: ui,i=0

ui,jj(y,t) = ∂2ui(y,t)/∂t

Velocity of S wave : vS = [/]1/2

Reciprocal theoremReciprocal theorem Elastic fields (u and v)

∫V{[]uj,ij(y,t) + ui,jj(y,t) - ∂2ui(y,t)/∂t2}*vi(y,t) dV

-∫Vui(y,t)*{[]vj,ij(y,t) + vi,jj(y,t) -∂2vi(y,t)/∂t2} dV.

Transform:

{[] uj,ij + ui,jj - ∂2ui/∂t2}*vi

= {vi*[[] uj,i+ ui,j]},j - vi*∂2ui/∂t2 - vi,j*{[]uj,i+ui,j}

DerivationDerivation

∫V{vi*[[] uj,i+ ui,j]},j dV- ∫Vvi*∂2ui/∂t2 dV

- ∫Vvi,j*{[] uj,i+ ui,j}dV -∫V{ui*[] vj,i+ vi,j]},j dV

+ ∫Vui*∂2vi/∂t2dV +∫Vui,j*{[] vj,i+ vi,j}dV

= ∫Svi*{[]uj,i+ ui,j]nj dS -∫Sui*{[]vj,i+ vi,j]nj dS

Relation : vi*∂2ui/∂t2 = ui*∂2vi/∂t2

Volume integral to Surface integralVolume integral to Surface integral ∫V{[+]uj,ij(y,t) + ui,jj(y,t) - ∂2ui(y,t)/∂t2}*vi(y,t) dV

- ∫Vui(y,t)*{[+]vj,ij(y,t) + vi,jj(y,t) -∂2vi(y,t)/∂t2} dV.

= ∫S [vi(y,t) *ij(y,t)nj  

- ui(y,t)*{[+]vj,i(y,t)+ vi,j(y,t)}nj]dS

ResultResult ∫V{[]uj,ij(y,t) + ui,jj(y,t) - ∂2ui(y,t)/∂t2}*vi(y,t) dV

- ∫Vui(y,t)*{[]vj,ij(y,t) + vi,jj(y,t) -∂2vi(y,t)/∂t2} dV.

= ∫S [vi(y,t) *ij(y,t)nj

- ui(y,t)*{[]vj,i(y,t)+ vi,j(y,t)}nj]dS

Note : {[]uj,i+ ui,j]nj = ij nj

Green’s FunctionsGreen’s Functions Green’s functions correspond to solutions of

[]Gkj,ij(x,y,t)+Gki,jj(x,y,t)-∂2Gki(x,y,t)/∂t2

= ki(x-y)(t)

a solution of displacement at x in the xk direction at time t due to an impulse at y in the xi direction at time t = 0. In the definition, no boundary conditions are taken into account.

Solutions of Green’s functionsSolutions of Green’s functions Green's functions in a half-space [Ohtsu & Ono 1984]

and an infinite plate [Pao & Ceranoglu 1981] are also available, taking into the boundary conditions. In these cases, however, solutions are obtained numerically, because some numerical integrations along the ray paths are requisite for computation.

In the case of a finite body, Green's functions can be obtained only by such numerical methods as the finite difference method (FDM) [Enoki, Kishi et al. 1986] and by the finite element method (FEM) [Hamstad, O’Gallagher et al. 1999].

Green’s function of the 2Green’s function of the 2ndnd kind kind

Traction : ti = ij nj ={[]uj,i+ ui,j]nj

The traction associated with Green's function

Tki(x,y,t) = {[] Gkj,i(x,y,t)+ Gki,j(x,y,t)}nj

= Gkp,q(x,y,t) Cpqij nj

Cpqij = [pqij + piqj + pjiq.

Setting vi =Gki

-∫Vui (y,t)*{[] vj,ij(y,t) + vi,jj(y,t)- ∂2vi(y,t)/∂t2}}dV  

=∫S [vi(y,t) *ij(y,t)nj  

- ui(y,t)*{[]vj,i(y,t)+ vi,j(y,t)}nj]dS

-∫Vui (y,t)*{- ki(x-y)(t)}dV = uk(x,t)

= ∫S [Gki(x,y,t)*ij(y,t)nj - ui(y,t)*Tki(x,y,t)]dS.

= ∫S [Gki(x,y,t)*tj (y,t)- ui(y,t)*Tki(x,y,t)]dS.

All solutions in elastodynamics are mathematically formulated as,

Dynamic displacements uk(x,t):AE wave

= ∫S[Gki(x,y,t)*ti(y,t) - Tki(x,y,t)*ui(y,t)]dS

This is a theoretical representation of AE wave

Theoretical TreatmentAE Waves due to Point ForceTheoretical AE WaveformsDeconvolution Analysis

IntroductionIntroduction In 1970’s, it was first demonstrated that AE waves

were elastic waves, which could be synthesized theoretically.

The famous paper by Breckenridge dealt with Lamb’s problem [Breckenridge & Greenspan 1981].

Later, the generalized theory of acoustic emission (AE) was established on the basis of elastodynamics [Ohtsu & Ono 1984].

Caution - 1Caution - 1 In a thin plate of steel, because the thickness of the

plate is often shorter than the wavelengths, the wavelengths of propagating waves are longer than the thickness. Thus, diffracted and dispersive waves are generated as guided waves.

Although it is also possible to analyze guided waves theoretically, they are no longer related with source dynamics.

To deal with source mechanisms, P wave and S waves are necessarily detected and analyzed.

AE waves due to a point forceAE waves due to a point forceHistorically, theoretical treatment started with

AE waves due to an applied force. In those days, because an experiment could be done easily, many people attempted to detected AE waves due to an applied force and attempted to apply experimental results to source characterization [Pao 1978].

Unfortunately, most papers were far from successful on source characterization, because the generalized theory was not understood clearly.

Caution - 2Caution - 2The basic concept of the generalized theory

results from the separation of AE waves due to the applied force and due to cracking.

If the source is different, the equation is also different.

Do not detect elastic waves due to bombing, if you are interested in earthquakes.

AE wave due to a forceAE wave due to a force Dynamic displacements: AE wave

uk(x,t)= ∫S[Gki(x,y,t)*ti(y,t) - Tki(x,y,t)*ui(y,t)]dS

On surface S tj(y,t) = f(t)j(y-y0) and uj(y,t) = 0

ui(x,t) = ∫Gij(x,y,t)* f(t)(y-y0)ej = Gij(x,y0,t)* fj(t)

the governing equation in the case that the governing equation in the case that dynamic force dynamic force ff(t) is applied at point (t) is applied at point yy00

on the specimen.on the specimen. In the case that force fj(t) is the step function h(t)ej,

ui(x,t) = Gij(x,y0,t)*h(t)ej= Gij(x,y0,t)*∫(t)dt ej

=∫Gij(x,y0,t)*d(t)dt ej=∫Gij(x,y0,t)ejdt= GijH(x,y0,t)ej  

GijH is Green’s function due to the step-function force,

GijH(x,y,t)/dt = Gij(x,y,t)

Green’s functions for Lamb problemGreen’s functions for Lamb problem

Gij(x,y,t) is an elastodynamic solution of the displacement in the xi direction at point x and at time t due to a delta-function force in the xj direction at point y and at time t = 0.

Green’s functions have to be computed numerically except for an infinite space where analytical solutions are known.

Lamb’s solutions corresponds to solutions in a half space.

Empirical Green’s function GEmpirical Green’s function GijijHH due to due to

the step-function force,the step-function force,

Step-function force:

Glass-capillary break

(Breckenridge)

Pencil-lead break

(Hsu)

Step-function force by Step-function force by a pencil-lead breaka pencil-lead break

Standard force of AE for calibration

Force

Time 0

Empirical Green’s functions GEmpirical Green’s functions GijijHH((xx,,yy,t),t)

In a practice, a pencil-lead break generates a step-function force. Therefore, AE waves detected by a capacitive sensor or a displacement sensor calibrated are equivalent to Gij

H.

This is a reason why many researchers mistakenly applied AE waves due to a pencil-lead break to source characterization. From the generalized theory, it is readily derived that AE waves due to the applied force are mathematically different from AE waves due to cracking.

Lamb’s problemLamb’s problem In elastodynamics as well as in seismology, the

problem where a force is applied in a half space is called Lamb’s problem.

This is because the problem was first solved by H. Lamb [1904]. Then, C. L. Pekeris published famous results Lamb’s solution due to a surface pulse [1955a] and a buried pulse [1955b].

A general solution suitable for numerical computation was comprehensively reported by L. R. Johnson [1974]. The computational code is already available in the literature [Ohtsu & Ono 1984].

Elastic wave is detected at a Elastic wave is detected at a surface due to a surface pulse surface due to a surface pulse

or a buried pulseor a buried pulse

Lamb’s solution for a buried pulseLamb’s solution for a buried pulse In the case that the

force is applied in the x3-direction and the displacement u3 in the x3-direction is observed,

u3(x,t) = GH33(x,y0,t)

Lamb’s solutions for a buried pulseLamb’s solutions for a buried pulse

Lamb’s solution for a surface pulse Lamb’s solution for a surface pulse Poisson’s ratios n = 0.25 and 0.3

Famous Breckenridge’s experiment Famous Breckenridge’s experiment [1982][1982] using a capacitance sensor due to using a capacitance sensor due to

a break of glass capillarya break of glass capillary

Experiment in a concrete blockExperiment in a concrete block

u3(x,t) = G33(x,y0,t)*f3(t)

A pulse signal was charged in a PZT element:f(t) = sin4(pt/Tr), 0<t<Tr

Lamb’s solutions detected by Lamb’s solutions detected by an AE sensoran AE sensor

Synthesized AE wavesSynthesized AE waves

uk(x,t)= ∫S[Gki(x,y,t)*ti(y,t)

- Tki(x,y,t)*ui(y,t)]dS

t(y,t) = 0

bi(y,t) = ui+(y,t) - ui

-(y,t).

Theoretical representation Theoretical representation

uk(x,t) = ∫F+[ - Tki+(x,y,t)*ui

+(y,t)]dF +

∫F-[- Tki-(x,y,t)*ui

-(y,t)]dF-.

uk(x,t) = -∫F+Tki(x,y,t)*[-ui

+(y,t)]]dF +∫F-[-Tki(x,y,t)*[ui

-(y,t)]

= ∫FTki(x,y,t)*[ui+(y,t) - ui

-(y,t)]dF

=∫FTki(x,y,t)*bi(y,t)dF.

=∫F Gkp,q(x,y,t) Cpqij nj*bi(y,t)dF

= Gkp,q(x,y,t)*S(t) Cpqij njli∫F b(y,t)dF

= Gkp,q(x,y,t)*S(t) Cpqij njliV,

Splitting test of concrete plateSplitting test of concrete plate

AE wave due to a tensile crackAE wave due to a tensile crack

Micrograph of debondingMicrograph of debonding

Synthesized AW waveSynthesized AW wave

Deconvolution AnalysisDeconvolution Analysis

inversely, the source-time function S(t) can be computed. The procedure is named the deconvolution analysis [Wadley & Scruby 1981]. Conventionally the source characterization of AE implies this procedure.

uk(x,t) = Gkp,q(x,y,t)*S(t) Cpqij njliV

S(t) = uk(x,t)// Gkp,q(x,y,t)Cpqij njliV

Results of deconvolutionResults of deconvolution

[Homework Nr. 2] reply to e-mail: mohtsu@ethz.ch[Homework Nr. 2] reply to e-mail: mohtsu@ethz.ch

 

Short Course 101-1111-00L: Fundamentals and Applications of Acoustic Emission by Prof. Masayasu Ohtsu

October 11

2-1 From a statistical relation in seismology, mkaan )( ,

where a is the amplitude, and n(a) is the number of earthquakes of the amplitude from a to a+da. Derive the amplitude distribution,

Log N(A) = – b Log A.

Here N(A) is the number of earthquakes up to the amplitude A, as A

daanAN0

)()(

2-2 From the governing equation of elastodynamics: Navier’s equation; [+]uj,ij(y,t) + ui,jj(y,t) = ∂2ui(y,t)/∂t. derive the velocities of P wave and S wave.