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Experimental validation of an analytical method to predict lamb wavescattering from a discontinuityTo cite this article: Mohammad Faisal Haider et al 2019 Smart Mater. Struct. 28 015012

 

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Experimental validation of an analyticalmethod to predict lamb wave scattering froma discontinuity

Mohammad Faisal Haider1 , Banibrata Poddar2 and Victor Giurgiutiu1

1Department of Mechanical Engineering, University of South Carolina, Columbia, SC, United States ofAmerica2 Intelligent Automation, Inc., Rockville, MD, United States of America

Received 16 July 2018, revised 25 September 2018Accepted for publication 17 October 2018Published 23 November 2018

AbstractThis paper presents an experimental validation of an analytical method called complex modeexpansion with vector projection (CMEP), which is used to calculate the scattering coefficients(amplitude of the out-of-plane velocity) of Lamb wave modes from geometric discontinuities.For a test case, a plate with a thickness step change type geometric discontinuity is considered inthis paper. The scattered wave fields from the discontinuity are expanded in terms of complexLamb wave modes with unknown scatter coefficients. These unknown coefficients are obtainedby projecting the stress or displacement boundary conditions on the displacement or stressboundary conditions utilizing the power expression. In the analytical analysis, complex-valuedscatter coefficients are calculated with frequency-thickness product from 50 to 1500 kHz mm forA0 incident wave. A parametric study was conducted using CMEP to find the optimized stepdepth ratio for the experiment. For incident A0 mode at step depth ratio of 0.6, the scatteringcoefficients of reflected and transmitted S0 modes are maximum. A plate of thickness 4.86 mmwith a step depth ratio of 0.6 was chosen for experimental study. Long piezoelectric wafer activesensors (PWAS) were used to create straight crested Lamb wave modes. Antisymmetric Lambwave mode selectively excited by using two PWAS in out of phase on opposite sides of theplate. Scanning laser Doppler vibrometer was used to measure the out-of-plane velocity ofscattered Lamb wave fields on the plate. Scatter coefficients were calculated from Fouriertransform of the time domain signal. The obtained experimental results agree well with theCMEP analytical predictions.

Keywords: Lamb waves scattering, analytical method, complex roots, Rayleigh–Lamb equation,straight crested wave, laser Doppler vibrometer

(Some figures may appear in colour only in the online journal)

1. Introduction

Cost effective and computational efficient structural healthmonitoring (SHM) and non-destructive evaluation (NDE) meth-ods are in urgent need of proper damage detection in practicalapplications. In general, damage detection techniques can bedivided into two groups: (a) active damage detection: energy isimparted to the structure using transducers to create elastic waves[1, 2] and (b) passive damage detection: detection of elasticwaves, generated due to rapid release of energy from damage orimpact [3, 4]. In active damage detection system, methods of

predicting the scattered Lamb wave modes from different typesof damage has been an interesting research topic for decades[5–8]. Lamb wave interacts with the structural discontinuity andresulting scattered Lamb wave modes as well as mode conver-sion. Scattered Lamb wave modes and mode conversion are thekey factors for characterizing damage in a structure.

Finite element method (FEM), the boundary elementmethod (BEM) are the popular methods for predicting scatterLamb waves from damage or discontinuity [9–12]. Manyresearchers also used a semi-analytical finite element (SAFE)technique for Lamb wave propagation problems [13, 14].

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However, FEM, BEM or SAFE require extensive computa-tional effort and computational speed associated with com-putational cost. In this paper, the CMEP approach is presentedfor calculating Lamb wave scatter coefficients. The scatteringcoefficients from a discontinuity depend on the frequency, thegeometry of the discontinuity and plate thickness. In theCMEP method, the scattered wave fields are expanded interms of complex Lamb wave modes with unknown scattercoefficients. These unknown coefficients are obtainedboundary-matching process with vector projection utilizingthe power expression. CMEP is a highly computational effi-cient and accurate analytical method for predicting Lambwave scattering [15, 16]. Many other researchers have alsoexplored an analytical approach to solve the scattering pro-blem [5, 15, 17–19]. One of the main challenges of thescattering problem is to satisfy the thickness dependentboundary conditions at the discontinuity.

A damage detection algorithm for identifying rectangularnotch parameters in a stepped waveguide using Lamb waveswas presented by Ghadami et al [20]. Scattering of shearwaves with the discontinuity in a plate could be useful fordetecting damage. Pau et al [21] investigated the interactionof the SH0 mode with discontinuity in plate waveguides. Thescattered SH0 wave fields were evaluated using an analyticaltechnique, and the obtained results were compared with thefinite element model [21]. Vien et al [22] presented a com-putational study of scattering of the symmetrical Lamb wavemode from a small edge crack in an isotropic plate. Fromme[23] predicted the scattered wave field directionality patternfor an incident low-frequency A0 Lamb wave mode from 3Dfinite element simulations and verified from experimentalmeasurements. On a similar aspect, Lie et al [24] employed athree-dimensional finite element model to study the incre-mental scattering of the antisymmetric (A0) Lamb wave modefrom notch emanating from through-holes.

The validation of any analytical technique is essentialwhen it comes to real-life applications. Alleyne et al [25]

studied the interaction of various types of notches with astraight crested Lamb wave using FEM and compared it withexperiments. The FEM analysis was done in 2D assumingstraight crested waves in plane strain condition. However, forexperimental validation, they assumed a circular crestedwavefront as straight crested at a distance from the transducer.A similar study was also performed by Lowe et al [26] wherethe prediction was made assuming a straight crested Lambwave, but the creation of straight crested waves has not beenconfirmed. Benmeddour et al studied the mode conversion ofS0 and A0 modes at symmetric and antisymmetric notches ina plate [27]. Santhanam et al [28] presented an experimentalvalidation of an analytical model using circular crested Lambwaves to predict the oblique reflection of straight crestedLamb waves. However, the creation of straight crested Lambwaves in a finite plate using finite transducers is quite chal-lenging and has not been reported in the literature. In thispaper, an experimental procedure is presented to generatestraight crested Lamb waves in a finite plate with finitetransducers. This experiment aims to validate the predictionsof CMEP, particularly for a Lamb wave mode interacting witha thickness step change.

This paper is organized as follow: section two presents ananalytical model based on the physics of the Lamb wavemodes to predict the scattering of Lamb wave from the step-like discontinuity. This section also presents the convergencestudy. Section three presents an experimental method tovalidate theoretical models predicting Lamb wave scatteringfrom a discontinuity. This section also discusses the detailedexperimental procedure. Finally, section four presents conclu-sions from this research and recommendation for future work.

2. Analytical formulation

The cross-section of a plate with step is shown in figure 1.The incident wave field is propagating in the +ve x-direction.

Figure 1. Schematic of Lamb wave interaction with stiffener on a plate.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

Upon interacting with the discontinuity, reflected wave fieldsand transmitted wave fields would be propagated in –ve x-direction and +ve x-direction respectively. As shown infigure 1, the discontinuity is located at x=x0. The platethickness in region 1 and region 2 is h1 and h2 respectively.The step depth of the plate is h h h .1 2 1-( ) For time-harmonicincident wave field, i.e.

u u e

e . 1

ij

ji x t

ij

ji x t

11

1

11

1

j

j

å

ås s

=

=

x w

x w

=

¥+ -

=

¥+ -

( )

( )

( )

The scattered wave field is the summation of the reflectedwave field (region 1) and the transmitted wave field (region 2).The scattered wave field can be written as

u C u e C u e

C e C e , 2

sj

jF

s ji x t

jB

s ji x t

sj

jF

s ji x t

jB

s ji x t

12 1

12 1

j j

j j

å

ås s s

= +

= +

x w x w

x w x w

=

¥+ - - -

=

¥+ - - -

( )

( ) ( )

( ) ( )

( ) ( )

where, u

and s is the displacement and stress vectors ofscattered wave fields; ξ is the wavenumber and w is the fre-quency. The subscript ‘i’ stands for incident waves and ‘s’stands for scattered and trapped waves. The subscript ‘j’represents different wave modes. The coefficient Cj

Bs is the

unknown amplitude of the of backward propagating (reflected)Lamb waves in region 1, whereas the coefficient Cj

Fs is the

unknown amplitude of forward propagating Lamb waves inregion 2. The amplitudes, Cj

Fs of these modes, must be

determined by boundary conditions using a vector projectionutilizing the power expression.

The plate should have zero-stress boundary condition atthe top and bottom surfaces

x x yh

x x y hh h

x x yh

x x y hh h

;2

,

;2

,2

0

;2

,

;2

,2

0 3

yy

xy

01

0 21 1

01

0 21 1

s

t

< =

> = - -=

< =

> = - -=

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

( )

The boundary conditions for the step change are shownin figure 2. The displacement boundary condition at thestep is

u u x x, at . 41 2 0= = ( )

The traction boundary conditions are

tt

x xx x

x xx x

0,

,atat

0,,

atat

. 5

12

0

0

12

0

0s

s

=-

==

===

⎧⎨⎩

⎧⎨⎩ ( )

Equations (4) and (5) represent the thickness dependentboundary conditions at the stiffener. CMEP formulationincorporates the average power flow associated with thescattered wave fields through boundary condition. For vectorprojection using power flow, the stress boundary conditionsare projected onto the conjugate displacement vector space(region 1), and the displacement boundary conditions areprojected onto the conjugate stress vector space (region 2) ofthe complex Lamb wave modes. To take advantage of theorthogonality of stress and displacement mode shapes ofLamb wave modes, u conj uj j1 1=

( ) is chosen as the projec-tion vectors for the stress boundary conditions and

conjj j2 2s s= ( ) as the projection vectors for the displace-

ment boundary conditions. The final form of the projectedboundary conditions are

u u dz

u u u x x

u dz u dz

u u x x

0

, , ;

, , ; . 6

b

j

S S j i j

h

j

b

j

S S j j

0

2

1 2 2

2 1 2 1 2 0

0

2

1 10

2

2 1

2 1 1 1 1 0

ò

ò ò

s

s s

s s

s s s

- =

á - ñ = á ñ =

=

á - ñ = á ñ =

( )

( )

( ) ( )

Equation (6) leads to a set of linear equations expressedas follow

A BE F

C

C

DG

Q C R . 7

j j

Fs

Bs j

j

j j j j

2 22 1

2 1

2 2 2 1 2 1

--

=

=

´´

´

´ ´ ´

⎪

⎪

⎪

⎪

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

⎫⎬⎭ { }

[ ] { } { } ( )

Equation (7) contains 2j algebraic equations with2j unknowns. Here, [Q] and {R} are known entities andC{ } contain the 2j unknowns. The unknown amplitudes ofthe scattered Lamb wave modes are to be solved asfollow

C Q R . 8j j j j2 1 2 21

2 1=´ ´-

´{ } [ ] [ ] ( )

2.1. Results and discussion

To implement the CMEP method, the complex modes ofLamb waves need to be included in the modal expansion.Therefore, the first step is to obtain all possible roots of theRayleigh–Lamb equation including an infinite number ofcomplex roots. Equation (9) are the Rayleigh–Lambequations for two basic wave modes: symmetric and

Figure 2. Boundary conditions at the stiffener interfaces.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

antisymmetric.

d d

d d

d d

d d

cos sin

4 sin cos 0 Symmetric

sin cos

4 cos sin 0 AntiSymmetric . 9

S P S

P S P S

S P S

P S P S

2 2 2

2

2 2 2

2

x h h h

x h h h h

x h h h

x h h h h

-

+ =

-

+ =

( )( )

( )( ) ( )

Here, ξ is the wavenumber; d is the half thickness of the plate;

ω is the circular frequency; ;P c2

P

2

2h x= -w ;P c2

S

2

2h x= -w cP,

cS are pressure and shear wave speeds respectively.Equation (9) can be expressed in terms of the dimen-

sionless wavenumber K dx= and the dimensionless fre-quency d cSwW = as

f K K a a b

abK a b

f K K a b a

abK b a

, sin cos

4 cos sin 0 Symmetric

, sin cos

4 cos sin 0 Antisymmetric , 10

S

A

2 2 2

2

2 2 2

2

W = -

+ =W = -

+ =

( ) ( )( )

( ) ( )( ) ( )

where, a K b K, ,2 2 1 2 2 2 2 1 2k= W - = W -( ) ( ) K =K iK ,real imag+ c c .P Sk =

At any frequency Ω, equation (10) has a large numberof roots of the complex plane. The roots are categorized as(a) real Kreal( ) roots corresponding to the forward propagatingLamb wave modes, (b) imaginary Kimag( ) roots correspondingto the evanescent modes and (c) complex roots K( ) corresp-onding to exponentially decaying amplitude wave modes[16]. Figure 3 shows all the complex roots of Rayleigh–Lambequations.

As a test case, an aluminum 6061 plate with E =68.9 GPa, 2700 kg m ,3r = - 0.33n = was considered. Theplate thickness in region 1 and region 2 are h1=4.86 mmand h2=1.94 mm respectively. Therefore, the depth ratio ofthe step becomes R h h h 0.6.2 1 1= - = For numericalanalysis, it needs to determine the maximum number of rootsof the Rayleigh–Lamb equation to be considered whichshould give a reasonably accurate result. A0 mode was

considered as the incident Lamb wave modes because A0mode has higher out-of-plane displacement amplitude than S0mode. The amplitude of the out-of-plane component would bemeasured experimentally using SLDV for experimental vali-dation. Frequency thickness product from 50 to 1500 kHz mmis chosen for its relevance to practical applications.

Figure 4 shows the convergence study for the amplitudesof the first two modes of scattered Lamb waves, S0 and A0.From the figure, it can be seen that by considering 27 modesin the modal expansion ensured the convergence with lessthan 0.5% error.

Lamb waves exhibit velocity dispersion; that is, theirvelocity propagation depends on the frequency and thickness.Therefore, it is convenient to use the frequency-thicknessproduct. Therefore, figure 5 shows the scattered wave (S0 andA0 modes) amplitudes and phase angle of out-of-plane

Figure 3 Complex roots of the Rayleigh–Lamb equation for v=0.33.

Figure 4. Convergence of amplitudes (uy) scattered Lamb wavemodes for incident A0 mode over a wide frequency-thicknessproduct from 50 to 1500 kHz mm.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

displacement with frequency-thickness ( f−h1) product forincident A0 modes.

From the figure 5(a), it can be seen that the amplitude ofthe A0 transmitted wave decreases with increasing frequencywhereas, the amplitude of A0 reflected, S0 transmitted, andS0 reflected increases with increasing frequency. The out-of-plane components corresponding to S0 wave mode issmaller than A0 mode, which is expected due to the nature ofstructural transfer function [29]. The phase angle of theout-of-plane displacement also changes with frequency(figure 5(b)).

3. Experiments

This section presents a detail of the experimental procedure tovalidate the CMEP results.

3.1. General experimental procedure

There are several major challenges for determining the scattercoefficients experimentally using straight crested Lambwaves. The following key factors should be considered togenerate straight crested Lamb wave successfully and todetermine the scatter coefficients experimentally:

a. Determining specimen geometry.b. Eliminating the boundary reflections.c. Creating a straight crested Lamb wave mode.d. Tuning symmetric or antisymmetric Lamb wave mode

selectively.e. Measuring the scattered wave field with sufficient

accuracy.

3.2. Determining the specimen geometry

To optimize the specimen geometry the predicted results ofthe scatter coefficient from CMEP was used for the experi-ments. The focus of the experiment was capturing the modeconversion of Lamb waves due to the step-like discontinuity.The scattered wave amplitudes at various depth ration werepredicted using CMEP (figure 6). Figures 6(a) and (b) showthat for incident A0 at depth ratio of 0.6 the scattering coef-ficients of reflected and transmitted S0 modes are maximum.Therefore, this depth ratio was chosen for the experiment. Forsafe handling, a small plate thickness after the step wasavoided; a plate of thickness 4.86 mm was considered. Thematerial for the plate was chosen as aluminum 6061 for goodmachinability.

In a pitch-catch wave propagation experiment, for reli-able measurement of scattered waves, the receiver should beat a sufficient distance from the source for separation ofsymmetric and antisymmetric modes. To predict the requireddistance, an existing analytical tool was used to simulatestraight crested Lamb waves (figure 7). The figure illustratesthat the sensors need to be placed at least 300 mm away fromthe step to detect the reflected wave modes and 250 mm forthe transmitted wave modes. A sufficient distance is alsoneeded between the edge of the plate and the actuator andsensors. There is another way to separate the modes usingfrequency-wavelength analysis from the time domain signalusing a double Fourier transform. This method could avoidthe unnecessary larger dimension of the plate, and theexperiment could be done within a confined space. However,in this experiment, a larger dimension of the plate was chosen.Considering all of the requirements, a 1220 mm× 305 mm×4.86 mm plate was selected for the experimental study.Figure 8 shows the plate after machining a step using avertical milling machine.

Figure 5. (a) Normalized amplitude and (b) phase of uy displacement for incident A0 mode (depth ratio R h h h 0.62 1 1= - = ).

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

3.3. Eliminating boundary reflections

To eliminate boundary reflections modeling clay as the dampingmaterial was applied on the surface of the plate around the edge.As shown in figure 9(a), the thickness of the modeling clay wasvaried starting from zero thickness to about 12mm thick at theedge. This was done so that the waves do not reflect from thebeginning of the clay boundary due to a sudden change inboundary conditions. Figures 9(b)–(d) shows the variation of theabsorbing clay boundary thickness in the real specimen.Figure 9(b) shows the entire plate after application of theabsorbing boundaries all around the specimen.

3.4. Creating a straight crested wave front

Three 40 mm×5 mm×0.2 mm piezoelectric wafer activesensor (PWAS) were bonded in a line on top and bottom

surface of the plate to create a long line source. However,due to the edge effect, the wave crest may not stay straightat the ends of a line source. A 3D FE model of a similarplate was created for understanding the straight crestedwavefront using long PWAS. For FEM simulation, a similarmodel as experimental setup was created as shown infigure 10(a). In FEM simulation, the size of the PWASwas kept same, and a non-reflecting boundary was alsoapplied.

Figure 10(b) shows the top view of the actual specimen.The non-reflecting boundary absorbed the curved crestedwaves produced around the corners of the PWAS. This resultsin a straight crested wavefront after propagating less than100 mm distance (figure 11(a)). In the experiment, threePWAS transducers were excited in phase and measured thewave field using SLDV. Figure 11(b) shows that the

Figure 6. CMEP results for variation of scattering coefficients of Lamb wave modes with frequency and depth ratio for incident A0 mode(a) reflected S0 (b) transmitted S0 (c) reflected A0 (d) transmitted A0.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

successful generation of straight crested A0 Lamb wave modeexperimentally with absorbing clay boundaries.

Figure 12(b) shows the signal corresponding to thegenerated A0 mode after traveling different distances. A littledifference of the wave amplitude (less than 5% dropped inamplitude) can be observed between two signals due to thelow dispersion of the A0 mode at 175 kHz as evident fromfigure 12(a). The wave amplitude does not change with thedistance, which confirms a successful generation of a straightcrested Lamb wave mode channeled by absorbing clayboundary. Figure 12(b) also establishes that the clay boundarydoes not damp the channeled wave.

3.5. Tuning S0 and A0 mode separately

To generate symmetric or antisymmetric mode selectively,PWAS transducers on both top and bottom surfaces of theplate were bonded with careful alignment (figure 13). SincePWAS transfer function is sensitive to the bonding condition[29], it is difficult to get the same transfer function for bothtop and bottom PWAS. For this reason, both the PWAS were

excited with amplitude and phase adjustment. To achieve this,a Tektronics AFG3052C dual channel signal generator wasused to tune A0 mode using 175 kHz (850 kHz mm) 3.5 counttone burst signal. Figure 13, confirms the procedure men-tioned above resulted in excitation of only A0 mode.

3.6. Measuring scattered wave field

To measure the scattered wave field, an SLDV was used tomeasure the out of plane velocity (vy) at the surface of the plate(figure 14). SLDV measurement was done at 12.5MHz sam-pling frequency with averaging of 100 sample. An area of650×40mm was scanned using SLDV. The size of the meshgrid of the SLDV measurement is 0.5 mm×0.25mm. A lowpass filter of 1.5MHz and a high pass filter of 100 Hz was usedduring the measurement. SLDV measurement provides theopportunity to visualize the wave modes scattered by the step.Figure 15 shows that the scattered wave modes are easilyvisible in the recorded wavefield data. Also, it is important tonote that the amplitude of the scattered symmetric mode ismuch smaller than the amplitude of the antisymmetric mode.

Figure 7. Separation of S0 and A0 modes predicted by the analytical method.

Figure 8. Aluminum plate with the step.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

This is because the antisymmetric modes have much higherout-of-plane velocity than the symmetric mode.

Cv

v

u

u

u

u, 11s y

s

yin

ys

yin

ys

yin

w

w

w w

w w

w

w= = =

( )( )

( )( )

( )( )

( )

where indices s and in stand for scattered and incident wavemodes, respectively.

For incident and reflected modes, wave fields data wererecorded at 300 and 350 mm distance before the steprespectively. For transmitted modes, wave fields data wererecorded 300 mm distance after the step. Power amplifierswere used for excitation of higher amplitude incident Lambwave to get stronger scattered wave signals. Figure 16 showsthe incident, reflected and transmitted time domain signal byaveraging SLDV measurement across the width.

At this point, the portions of the signal corresponding todifferent modes were separated and converted the timedomain signal into the frequency domain signal using FFTalgorithm (figure 17). Then, the scattering coefficients Cs

were calculated using the equation (11). Also fromequation (11), it can be seen that either the velocity (vy) ordisplacement (uy) will result in the same scatter coefficient.

3.7. Comparison between CMEP and experimental results

Figure 18 shows the comparison between experimental resultsand analytical results with frequency-thickness values. Therange of frequency-thickness was chosen based on the ana-lytical results (figures 5 and 6). At low frequency, the ampl-itude of the scattering coefficient is very small for S0transmitted and S0 reflected modes. The amplitude of thescattered S0 mode increases with increasing frequency-thickness product. However, with increasing frequency-thickness value the amplitude of the scattered A0 modedecreases. Moreover, higher frequency-thickness may notsuitable for practical applications as the energy of the signaldissipates (signal attenuation) faster. Therefore, an optimizedfrequency-thickness range 650–1050 kHz mm was chosen forthe experimental study. For 4.86 mm thick plate 175 kHz(850 kHz mm) center frequency of the tone burst signal isgood enough to capture the amplitude of the scattered signalfrom 650 to 1050 kHz mm frequency-thickness range. Itshould be noted that for the tone burst signal the amplitude ofthe signal is dominant at the center frequency. The amplitudeof the signal diminishes as the frequency goes far away fromthe center frequency (figure 17). Therefore, the excitation

Figure 9. Creating absorbing boundary with modeling clay (a) schematic diagram (b) the left end of the plate after application of clay (c) theright end of the plate after application of clay (d) absorbing boundary around the plate.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

Figure 10 Top view of the specimen (a) schematic diagram (b) actual specimen.

Figure 11 Creating straight crested waves using non-reflecting boundary (a) FEM simulation (b) experimental result.

Figure 12 (a) Dispersion curve for 4.86 mm thick aluminum plate (b) experimental incident wave after traveling different distances.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

frequency should be chosen in such a way so that the fre-quency-thickness corresponding to the excitation frequencyof the tone burst signal is at the center of the desired fre-quency-thickness range.

From figure 18 shows that the experimental results are inagreement with the CMEP results. The maximum percentageof error was found 14.3%, 8.5%, 3.2% and 10.1% for

reflected S0, reflected A0, transmitted S0 and transmitted A0mode respectively. However, the percentage of error at thecenter frequency-thickness value was found 7%, 7.5%, 1.5%and 4% for reflected S0, A0, transmitted S0 and A0 modesrespectively. It can be inferred that the percentage of errordecreases as the frequency-thickness value shifts towards theexcitation frequency-thickness value. Therefore, multiple

Figure 13 Tuned A0 mode using top and bottom PWAS.

Figure 14 Experimental setup for out-of-plane velocity measurement using SLDV.

Figure 15 Experimentally measured scattered straight crested Lamb wave modes using SLDV.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

Figure 16. Experimentally measured time domain signal (averaged signal across the width from SLDV measurement) (a) incident andreflected waves and (b) transmitted waves modes.

Figure 17 FFT of the incident reflected and transmitted time domain signal.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

excitation signals with different center frequency wouldprovide better experimental results.

Measurement of scattering of Lamb waves is, in general,a challenging problem. From these figures, it can be observedthat the experimentally obtained scatter coefficients forreflected and transmitted A0 modes were consistently lowerthan the predicted ones. The CMEP predictions assumed nomaterial damping. Whereas, in reality, there is materialdamping and its effect is higher on A0 mode than the S0mode. Since the scattered waves were measured at a sig-nificant distance away from the step, this effect of dampingmay result in lower than the predicted amplitude of thereflected and transmitted modes, particularly for A0 mode.The signals at different locations are susceptible to the phaseshifts, and that may happen due to spatial misalignment of theSLDV measurement. Also, in this experiment, a 3.5 cycletone burst excitation signal was used, which is an additionalsource of estimation error because of the frequency depend-ence of the scattering coefficient. More cycles signal would

result in a narrower bandwidth, which is desirable for fre-quency purity as well as for causing less dispersion. However,because of the relatively short propagation distances for bothexperiments, 3.5 cycles were selected to maintain separationin time of the desired scattered signals. Step change may notbe exact verticle due to manufacturing constraint, which mayalso affect the estimation results.

However, all the four plots show the trends of the scat-tering coefficient predicted by CMEP are same as that of theexperimentally obtained results. The values of the scatteringcoefficients are enough close to the predicted values fromCMEP. The results shows that CMEP is a reliable tool topredict the scattering of the Lamb waves from geometricdiscontinuities. In comparison with other NDE or SHMtechnique, this analytical method capitalizing on Lamb wavesscattering that can offer faster, more accurate and cost-effective evaluation of different types of damage. Thismethod is useful for different types of damage to whichultrasonic Lamb waves are particularly sensitive including

Figure 18. Experiment versus CMEP result for variation of scattering coefficients of different Lamb wave modes for incident A0 mode(a) reflected S0(b) reflected A0 (c) transmitted S0 (d) transmitted A0.

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Smart Mater. Struct. 28 (2019) 015012 M F Haider et al

discontinuity, corrosion patch, debonding, cracking, etc.Damage identification technique using CMEP is envisionedto be a promising method in addition to traditional;NDE/SHM approaches.

4. Conclusion

In this paper, a cost-effective, reliable, computational efficientanalytical method called CMEP was demonstrated for pre-dicting Lamb wave scattering from a discontinuity. Thescattered wave fields are expanded in terms of complex Lambwave modes with unknown scatter coefficients. Theseunknown coefficients are obtained from the vector projectionof boundary conditions utilizing the power expression. Theanalytical results show that scattered coefficients depend onthe frequency and the geometry of the discontinuity.

An experimental procedure was also demonstrated tocreate straight crested Lamb wave modes using long PWAStransducer. Experimental results showed that it is possible tocreate straight crested Lamb wave modes using absorbingboundary along with long transducers. This method of gen-erating straight crested Lamb waves would be useful for manyexperimental investigation and validation of predictive mod-els. For this research, this method was used to validate thepredictions from CMEP for a step-like discontinuity in aplate. The obtained experimental results agree with the CMEPpredictions well. Scatter coefficients can be used to predict thescattered time domain signal. Time domain signal is useful forpractical applications as it is observable from an SHM/NDEsystem. Future work would be recommended to simulatescattered time domain signal using the scattering coefficient.Future work would also be recommended for using CMEP formore complex geometry, e.g. several or non-limited step-likechanges to calculate scattering coefficients.

Acknowledgments

Support from the Office of Naval Research # N00014-11-1-0271, Dr Ignacio Perez, Technical Representative, and Air ForceOffice of Scientific Research #FA9550-11-1-0133, Dr DavidStargel, Program Manager; are thankfully acknowledged.

ORCID iDs

Mohammad Faisal Haider https://orcid.org/0000-0003-4814-1673

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