Evaluation of fatigue damage using nonlinear guided waves

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 93.180.53.211

This content was downloaded on 03/10/2013 at 06:10

Please note that terms and conditions apply.

2009 Smart Mater. Struct. 18 035003

(http://iopscience.iop.org/0964-1726/18/3/035003)

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 18 (2009) 035003 (7pp) doi:10.1088/0964-1726/18/3/035003

Evaluation of fatigue damage usingnonlinear guided wavesChristoph Pruell1, Jin-Yeon Kim1, Jianmin Qu2 andLaurence J Jacobs1,2

1 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta,GA 30332-0355, USA2 G W Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,GA 30332-0405, USA

Received 14 July 2008, in final form 22 November 2008Published 27 January 2009Online at stacks.iop.org/SMS/18/035003

AbstractThis research develops an experimental procedure for characterizing fatigue damage in metallicplates using nonlinear guided waves. The work first considers the propagation of nonlinearwaves in a dispersive medium and determines the theoretical and practical considerations forthe generation of higher order harmonics in guided waves. By using results from the nonlinearoptics literature, it is possible to demonstrate that both phase and group velocity matching areessential for the practical generation of nonlinear guided elastic waves. Next, the normalizedacoustic nonlinearity of low cycle fatigue damaged aluminum specimens is measured withLamb waves. A pair of wedge transducers is used to generate and detect the fundamental andsecond harmonic Lamb waves. The results show that the normalized acoustic nonlinearitymeasured with Lamb waves is directly related to fatigue damage in a fashion that is similar tothe behavior of longitudinal and Rayleigh waves. This normalized acoustic nonlinearity is thencompared with the measured cumulative plastic strain to confirm that these two parameters arerelated, and to reinforce the notion that Lamb waves can be used to quantitatively assessplasticity driven fatigue damage using established higher harmonic generation techniques.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Ultrasonic techniques have been extensively used forinspecting and monitoring various engineering structures andcomponents, but generally only utilize the linear behaviorof the ultrasound. These linear techniques are effective indetecting discontinuities in materials such as cracks, voids, orinclusions, but are ineffective in monitoring damage prior tocrack initiation. In contrast, nonlinear ultrasonic techniquescan quantitatively detect and characterize plasticity drivenmaterial damage prior to the formation of micro-cracks. Fordamage types caused by plastic deformation such as fatiguedamage, nonlinear ultrasonic methods offer an especially highpotential to quantitatively track damage before the initiationof the first crack. The physical effect that is monitoredin nonlinear ultrasonic measurements is the generation ofhigher harmonic frequencies in an originally single frequencysignal. This higher harmonic generation may come from eitherlattice anharmonicity or microstructure defects. The latter is

typically the predominant source of higher order harmonics infatigue damaged metals. The magnitude of the second orderharmonic has been used to define the acoustic nonlinearityparameter, β . Theoretically, this β is an intrinsic materialparameter that describes the degree of nonlinearity of andfatigue damage in the material. Previous research [1–5] hasclearly demonstrated that the amount of cumulative damagein metal due to either monotonic plastic deformation, orcyclic fatigue is closely correlated with β as measured withthe second order harmonics. These studies also found thatnonlinear ultrasonic methods are indeed more sensitive todamage than any known method based on the measurementsof linear parameters such as wave speed and attenuation.

Guided ultrasonic waves, such as Lamb waves, haveunique advantages over other ultrasonic techniques that usebulk waves in that they can efficiently interrogate largeareas and geometrically complex or inaccessible components.Techniques based on guided waves have seen numeroussuccesses in detecting discontinuities and cracks over extended

0964-1726/09/035003+07$30.00 © 2009 IOP Publishing Ltd Printed in the UK1

Smart Mater. Struct. 18 (2009) 035003 C Pruell et al

distances in many different types of materials and structuralcomponents. Most guided wave techniques are based onlinear ultrasound. It is thus desirable to combine the largearea, long-range coverage of guided waves with nonlinearultrasonics to enable the quantitative assessment of materialdamage. Unfortunately, an accurate experimental realizationof nonlinear Lamb waves is generally difficult [6–10] due totheir dispersive and multi-mode nature. Deng and Pei [11]conducted nonlinear Lamb wave measurements in aluminumspecimens subjected to cyclic fatigue loading, using a stresswave factor (SWF) defined as the absolute magnitude of thesecond harmonic signal integrated over a certain frequencyrange as a measure of the acoustic nonlinearity. Rathersurprisingly, their SWF monotonically decreases as the numberof fatigue cycles increases; this is contrary to the trend ofincreasing acoustic nonlinearity with increasing plasticity thathas been seen in longitudinal wave measurements.

The current paper more thoroughly develops thetheoretical background for the propagation of nonlinear guidedultrasonic waves in a multi-mode dispersive medium, and thenuses a recently developed experimental procedure [12, 13] toquantify plasticity driven fatigue damage with nonlinear Lambwaves, relating these results to the level of plastic strain andremaining fatigue life.

2. Propagation of nonlinear waves in a dispersivemedium

Consider the second harmonic generation of elastic waves(a finite sinusoidal wavepacket) propagating in a weaklynonlinear, dispersive medium. In general, second harmonicwaves in such a medium are characterized as long, noise-likewave trains before and/or after the fundamental (primary) wavepulse. This behavior is due to dispersion; since the secondharmonic wavepackets are generated at different locations (andtimes) from the fundamental wave pulse, they will combine toform a continuously dispersed wave trail.

If the dispersive medium is a plate-like waveguide, thenthe amplitude of the second harmonic Lamb wave can grow(so it is cumulative) as the fundamental pulse propagatesonly when certain conditions are satisfied [6, 14–17]. Thefirst condition is synchronism, i.e., the matching of phasevelocities. Physically, synchronism is the condition thatenables the energy to transfer from the fundamental to thesecond harmonic mode on a continuous and instantaneousbasis. The equal phase velocity is called the ‘synchronousvelocity’. When the nonlinearity is weak, the second harmonicwave is generated and driven by the fundamental wave [6]and this action can be reinforced only when the secondharmonic wave’s phase changes exactly twice as fast as thefundamental wave’s (so that the two waves are synchronized).If the fundamental wave is a continuous time-harmonic wavetrain (a very long wavepacket), this condition is sufficientfor the second harmonic wave to grow as it propagates [15].However, when the length of the fundamental pulse isrelatively short in time, the second harmonic wavepacket willstart to separate from the fundamental one, and eventually thesecond harmonic wave will stop growing if the fundamental

and second harmonic waves do not have the same groupvelocity. Therefore, when the fundamental wavepacket isshort, in addition to matching the phase velocity, the groupvelocity of the fundamental and second harmonic waves needsto be the same in order for the second harmonic wave tobe cumulative. This additional requirement on matching thegroup velocity ensures that the energy transferred from thefundamental frequency (in the form of the second harmonic)stays within the same (second harmonic) wavepacket, thusaccumulating. These conditions have been extensivelydiscussed in the nonlinear optics literature for the purpose ofproducing a very short femtosecond laser pulse [14, 18–20]through the process of second harmonic generation. They aregenerally applicable to any type of dispersive media and canbe simultaneously achieved in a multi-mode dispersive mediasuch as a waveguide.

One last condition for the cumulative second harmonicgeneration in a waveguide (elastic plate) is non-zero powerflux from the primary mode to the second harmonic modeas noted in [6]. The power flux can be interpreted as theforce that is driving the second harmonic wave which comesfrom the primary wave due to the material nonlinearity. deLima and Hamilton [6] showed that for a plate with quadraticnonlinearity, no power flux can occur from a symmetric Lambmode to a non-symmetric Lamb mode, or vice versa from anon-symmetric to a symmetric Lamb mode. This condition canbe met by ensuring that the primary and the second harmonicmodes are of the same kind (symmetric or non-symmetric).

For a pair of synchronous Lamb modes that satisfiesthe condition of either matching group velocities or non-zeropower flux between them, the shape of their wavepackets maychange as they propagate, since their group velocity can bedifferent from their phase velocity. But they are not dispersivein that these two modes move in a single wavepacket. Note thatthe longitudinal and Rayleigh waves in an unbounded mediumsatisfy these conditions and their second harmonics propagatecumulatively. In principle, there are an infinite number of modepairs that satisfy these conditions in an elastic plate. This isespecially true in the high frequency range where dispersioncurves tend to be horizontal.

Figure 1 shows two mode pairs in both the phase andgroup velocity dispersion curves for a 1 mm thick aluminumplate—the mode pair (s1, s2) at normalized frequencies f h =3.57 and 7.14 MHz mm, where h is the plate thickness inmillimeters, with the synchronous velocity 6349 m s−1 [9, 12]and the mode pairs (s2, s4) and (a2, a4) at frequenciesf h = 5.08 and 10.16 MHz mm, with the synchronous velocity8080 m s−1 [7, 8, 11]. As can be seen in figure 1, the (s1, s2)mode pair satisfies both of the above conditions, while both(s2, s4) and (a2, a4) mode pairs have very different groupvelocities. Therefore, the second harmonic wave amplitudein this mode pair will saturate with propagation distance,making this mode pair unsuitable for long propagation distanceinterrogation; this prediction is observed in the results of Denget al [7, 8]. In practice, modal excitability is another factor tobe taken into account. A careful experimental and theoreticalexamination is needed to determine a mode pair that is bestin both generating (i.e. modal excitability) and identifying

2

Smart Mater. Struct. 18 (2009) 035003 C Pruell et al

(a)

(b)

Figure 1. Normalized dispersion curves for an aluminum plate:(a) phase velocity versus normalized frequency; and (b) groupvelocity versus normalized frequency.

the second harmonic wave amplitude. This study uses themode pair (s1, s2) as in [9, 10]. A practical considerationis the selection of well-separated modes to avoid spuriouscontributions to the second harmonic amplitude calculation;the second harmonic mode (s2) propagates fastest in thewavepacket, so it can be detected without being significantlyinfluenced by other non-synchronous second harmonic modesthat can potentially dominate the measured multi-mode signal.For the 1.6 mm thick aluminum plate used in this study, thefundamental frequency selected is 2.225 MHz.

3. Experimental procedure

3.1. Specimens and fatigue testing

Three dog-bone specimens (16 mm width and 430 mm length)are cut from the same 1.6 mm thick Al-1100-H14 plate toensure consistency in the initial material nonlinearity of allthree specimens. Figure 2 shows a schematic of the specimens.A sinusoidal axial stress load with a frequency of 1 Hz isapplied in a stress-control mode using a commercial load frame

(10 ton MTS machine with hydraulic grips) with a stressratio R = 0. The maximum stress amplitude is σmax =125 MPa, the minimum stress amplitude is σmin = 0 MPaand the mean (average) stress amplitude is σa = 62.5 MPa.The maximum stress is higher than the yield stress of thematerial (σyield = 97 MPa), meaning these are low cyclefatigue tests. The fatigue tests are interrupted at 5, 15 and 50cycles and the specimens are removed from the load frame inorder to conduct the nonlinear ultrasonic measurements. Thecumulative plastic strain is calculated as the change in lengthbetween two fixed points divided by its original length—twopoints in the gage length are marked and the distance betweenthem is measured with a digital micrometer when the specimenis unloaded and removed from the load frame. The numberof fatigue cycles to failure for each of the three specimensis 60, 89, and 45 cycles. Note that fatigue testing of metalswill have an inherent scatter in cycles to failure due to therandom nature of each specimen’s initial microstructure, andthe evolution of fatigue damage. Another source of scatterin this research is the removal from, and then return of, thespecimens to the load frame for the ultrasonic measurements.This process may have introduced some inconsistency in theloading conditions, leading to a shorter or longer fatigue life.However, the focus of this research is to show the existenceof a relationship between acoustic nonlinearity measured withLamb waves and plasticity driven fatigue damage, and notto provide a comprehensive data set that could be used forquantitative fatigue life prediction.

3.2. Nonlinear ultrasonic measurements

This study uses an improved version of a previously proposedexperimental technique [9, 10] which is essentially the sameas the one used in [12]. Figure 3 shows a schematic of theexperimental setup. A high voltage tone burst signal (with apeak-to-peak voltage 698Vpp with the load of the ultrasonictransducer) of 25 cycles at a frequency of 2.225 MHz generatedby a high power gated amplifier (RITEC RAM–5000 Mark IV)is fed into a narrowband ultrasonic transducer (Panametrics X-1055) having a center frequency of 2.25 MHz and an activeelement diameter of 12.5 mm. The transducer is coupled to aPlexiglas wedge which is designed to launch a specific Lambmode (s1) into an aluminum plate of thickness 1.6 mm. Thewedge is coupled to the specimen using light lubrication oil.For a more efficient generation and reception of the targetedmodes, the shape of the wedge has been improved—the shapeis modified in such a way that the propagation distance inthe wedge is minimized, and a spring retainer is used toapply a constant pressure on the transducer, which helpsachieve a consistent oil-coupling to the wedge. A narrowbandultrasonic transducer (Panametrics A-109S) with a 5 MHzcenter frequency and a 12.5 mm active element diameter isused in the receiving wedge. The receiving wedge transducersimultaneously (at one angle) detects both the s1 and s2modes since their phase velocities are equal. The piezoelectricresonant behavior of the wedge transducer receiver has theadvantage that it is more sensitive to the second harmonicLamb wave mode which is relatively weak, thus improving thesignal-to-noise ratio (SNR).

3

Smart Mater. Struct. 18 (2009) 035003 C Pruell et al

Figure 2. Schematic of fatigue specimen (all dimensions in mm).

Figure 3. Schematic of experimental setup.

As in previous work [5, 9, 12, 13], consider the ratioA2/A2

1 as a measure of the normalized acoustic nonlinearity,where A1 and A2 are the measured (uncalibrated) amplitudesof the fundamental and second harmonic waves, respectively.This ratio of A2/A2

1 is used as a measure of the acousticnonlinearity parameter relative to its initial value and isproportional to the absolute acoustic nonlinearity parameter, β .As was shown for Rayleigh waves [5], the acoustic nonlinearity(β) is an experimentally measurable material property thathas a universal form β = 8A2/(kx A2

1)C , where A2 is theamplitude of the second harmonic, A1 the amplitude of thefundamental frequency, k the wavenumber, x the propagationdistance and C is a correction factor which only dependson the wave type. C = 1 for longitudinal waves, andit has a more complicated form for Rayleigh waves [5].However, this correction factor is not known for Lamb waves.The ratio A2/A2

1 is the factor which contains the essentialinformation on the nonlinear wave propagation, so it is selectedas the normalized acoustic nonlinearity parameter without thecorrection factor. As noted in section 2, this study uses themode pair (s1, s2), so A1 is the measured amplitude value of thes1 mode at the fundamental frequency, 2.225 MHz, while A2

is the measured amplitude value of the s2 mode at the secondharmonic frequency, 4.45 MHz.

The dispersive and multi-mode nature of Lamb wavesintroduces difficulties in extracting the Lamb wave amplitudesat the fundamental (A1 at 2.225 MHz) and second harmonic(A2 at 4.45 MHz) frequencies from a measured time-domain signal. Because several different Lamb modes cancontribute to the amplitudes at the fundamental and secondharmonic frequencies, the ordinary Fourier spectrum willinclude spurious contributions from other modes and thus giveinaccurate results. Following the procedure described in detail

in [12], a measured time-domain signal is processed in thetime–frequency domain with the short time Fourier transform(STFT) [21] to obtain its spectrogram. This spectrogram isthe energy density of the STFT, and has proven to be effectivein resolving the individual mode contributions of a transient,multi-mode, Lamb wave time-domain signal. Use of thespectrogram enables a direct, simultaneous measure of boththe fundamental amplitude, A1 of the s1 mode, and the secondharmonic amplitude, A2 of the s2 mode.

Figure 4(a) shows a typical transient time-domain Lambwave signal measured in an undamaged specimen. Thiswaveform, which is an average of 1000 signals to improve theSNR, is extremely complicated because it contains multiple,closely spaced, dispersive modes. As a result, the arrival timesof the individual modes are nearly impossible to determinein the time domain. Instead, consider the STFT of thissignal. As discussed in [12], the window size of the Fouriertransform has to be small enough to effectively separate thedifferent mode contributions when the propagation distance isrelatively short—the multiple modes are not well separatedat short propagation distances. Specifically, this study uses a64 point Hanning (Hann) window (with a 60 point overlap)and note that each time-domain signal has a record length of15 000 points with a sampling rate of 25 MHz. Figure 4(b)is the spectrogram of this time-domain signal together withthe theoretical group velocity curves, antisymmetric modes assolid lines, and symmetric modes as dashed lines. In contrastto figure 4(a), the arrival times of the different modes can beidentified in this spectrogram. Note that the time delay in thewedge is taken into account to get the theoretical dispersioncurves with the correct arrival times. The frequency spreadaround 2.225 MHz is attributed to the insufficient dampingof the source transducer. The fundamental (2.225 MHz)and second harmonic (4.45 MHz) frequencies are shown asdotted vertical lines, and figure 4(c) shows slices at thesetwo frequencies. The amplitudes of the fundamental andsecond harmonic modes, A1 and A2, are determined inthese two slices, and are used to calculate the normalizedacoustic nonlinearity parameter, A2/A2

1. Finally note thatthere is no discernible visual difference between time-domainsignals (or spectrograms) measured in damaged or undamagedspecimens; these differences are only made obvious throughthe calculation of A2/A2

1.

4. Experimental results and discussion

First consider the plastic strain results; figure 5 shows themeasured cumulative plastic strain versus the number offatigue cycles. All three specimens show similar trends inplastic strain, with a rapid increase in cumulative plastic strain

4

Smart Mater. Struct. 18 (2009) 035003 C Pruell et al

(a)

(b)

(c)

Figure 4. Results from nonlinear ultrasonic measurements:(a) typical time-domain signal; (b) resulting spectrogram withdispersion curves; and (c) slices at the fundamental and secondharmonic frequencies as a function of time.

during the first 15 cycles, followed by a slower increase. Thiscumulative plastic strain is a direct measure of fatigue damage,and it will be used to benchmark and interpret the nonlinearultrasonic results.

Figure 5. Measured cumulative plastic strain versus fatigue cyclenumber.

Most metals will undergo hardening or softening undercyclic loading. With an increasing number of cycles,the dislocation density will increase, causing the stress–strain behavior to change during fatigue. Many aluminumalloys harden under cyclic loading due to fine precipitates.Following [22], the relationship between plastic strainamplitude and the number of cycles can be represented witha straight line in a double logarithmic plot. A calculation canbe made for Al-1100-H14 to predict the cumulative plasticstrain as a function of fatigue cycles for this material; figure 5includes these results as the calculated curve. Note that thiscyclic hardening behavior for aluminum is not surprising andis caused by the accumulation of dislocations in the latticestructure as well as the transformation of these dislocations topersistent slip bands (PSBs) [22].

Next consider the nonlinear ultrasonic results. Aseries of preliminary measurements are performed in the farfield to ensure that amplitude of the normalized acousticnonlinearity, A2/A2

1 increases linearly with propagationdistance. Demonstration of this cumulative behavior iscritical to ensure that what is being measured is the materialnonlinearity, and not simply instrumentation nonlinearity.These results, which are shown in figure 6 for two differentaluminum specimens, are used to benchmark the resultspresented in [9] and are an indication that the proposedexperimental procedure provides a direct measure of materialnonlinearity.

Figure 7 shows the normalized acoustic nonlinearityversus percentage of fatigue life, plus a best fit curveand the error bars associated with the nonlinear ultrasonicmeasurements. The error bars are determined by repeating thenonlinear ultrasonic measurements three times at each stage offatigue life—the initial (undamaged) state, and after 5, 15 and50 cycles. The wedges for both generation and detection arecompletely removed and reattached to the specimen for eachmeasurement. The percentage of fatigue life is calculated bydividing the cycle number by the number of cycles to failurefor that specific specimen.

5

Smart Mater. Struct. 18 (2009) 035003 C Pruell et al

Figure 6. Normalized acoustic nonlinearity versus propagationdistance.

Figure 7. Normalized acoustic nonlinearity versus per cent of fatiguelife: mean values, best fit, and error bars.

Figure 7 shows that the increase in normalized acousticnonlinearity is rapid during the initial stages of fatigue life,and then progresses more slowly towards the end. Thisbehavior is in agreement with previous results for fatiguedamage due to low cycle fatigue measured with both nonlinearlongitudinal [1, 2, 4] and Rayleigh [5] waves. These resultsfurther demonstrate that nonlinear ultrasonic measurements arevery useful in quantitatively characterizing the fatigue stateof a material, especially in the early stages of fatigue life.The fatigue damage causes an increase in acoustic nonlinearitymeasured with Lamb waves in a manner similar to longitudinaland Rayleigh waves. This implies that the dispersive natureof Lamb waves does not alter their interaction with fatiguedamage and that there is a fundamental relationship betweenfatigue damage and acoustic nonlinearity, independent of wavetype.

Figure 8. Normalized acoustic nonlinearity versus plastic strain.

Figure 8 compares the normalized acoustic nonlinearitywith measured plastic strain. As expected, these twoparameters are clearly related. This behavior further linksthe normalized acoustic nonlinearity to a specific materialbehavior, plastic strain, and reinforces the notion that Lambwaves can be used to quantitatively assess plasticity drivenfatigue damage. Previous models [23, 24] have theoreticallyrelated plastic strain to the generation of higher harmonics, andthe current results qualitatively confirm this relationship.

Finally note that damage in fatigue tends to be highlylocalized, and the increase in acoustic nonlinearity is smaller,as distance increases away from the damage site. Forexample, [1] shows that the acoustic nonlinearity measuredwith longitudinal waves at different locations in a fatiguedamaged specimen can vary by as much as 140%, dependingon spatial location. This indicates that acoustic nonlinearitystrongly depends on the volume of the specimen beingmeasured. However, Lamb waves propagate long distances andwill therefore provide results averaged over a relatively largevolume. This imposes some limitations on using Lamb wavesfor long-range inspection. For the practical implementationof this technique, one should consider the trade-off betweensensitivity and propagation (inspection) distance.

5. Conclusions

A piezoelectric transducer based ultrasonic measurementtechnique is developed to quantitatively assess fatigue damagein aluminum with nonlinear guided waves. The generation ofhigher order harmonics in guided waves is considered, and itis demonstrated that both phase and group velocity matching isrequired for the practical generation of nonlinear guided elasticwaves. The results of this research show that the normalizedacoustic nonlinearity measured with Lamb waves is related tofatigue damage in a fashion that is similar to the behavior oflongitudinal and Rayleigh waves. This normalized acousticnonlinearity is then compared with the measured cumulativeplastic strain to confirm the existence of a relationship between

6

Smart Mater. Struct. 18 (2009) 035003 C Pruell et al

these two parameters, and to reinforce the notion that Lambwaves can be used to quantitatively assess plasticity drivenfatigue damage. This is an important finding for practicalstructural health monitoring and life prediction applications inthat Lamb waves can be used to quantitatively assess plasticitydriven material damage within the framework of establishedhigher order harmonic generation techniques.

Acknowledgments

This work was partially supported by the National ScienceFoundation under CMMI-0653883 and Air Force Office ofScientific Research under contract number FA9550-08-1-0241.

References

[1] Cantrell J H and Yost W T 2001 Nonlinear ultrasoniccharacterization of fatigue microstructures Int. J. Fatigue23 S487–90

[2] Frouin J, Matikas T E, Na J K and Sathish S 1999 In situmonitoring of acoustic linear and non-linear behavior oftitanium alloys during cyclic loading NondestructiveEvaluation of Aging Materials and Composites III; Proc.SPIE 3585 107–16

[3] Barnard D J, Dace G E and Buck O 1997 Acoustic harmonicgeneration due to thermal embrittlement of Inconel 718J. Nondestruct. Eval. 16 67–75

[4] Kim J-Y, Jacobs L J, Qu J and Littles J W 2006 Experimentalcharacterization of fatigue damage in a nickel-basesuperalloy using nonlinear ultrasonic waves J. Acoust. Soc.Am. 120 1266–73

[5] Herrmann J, Kim J-Y, Jacobs L J, Qu J, Littles J W andSavage M F 2006 Assessment of material damage in anickel-base superalloy using nonlinear Rayleigh surfacewaves J. Appl. Phys. 99 124913

[6] de Lima W J N and Hamilton M F 2003 Finite-amplitudewaves in isotropic elastic plates J. Sound Vib. 265 819–39

[7] Deng M, Wang P and Lv X 2005 Experimental observation ofcumulative second-harmonic generation of Lamb-wavepropagation in an elastic plate J. Phys. D: Appl. Phys.38 344–53

[8] Deng M, Wang P and Lv X 2005 Experimental verification ofcumulative growth effect of second harmonics of Lamb wavepropagation in an elastic plate Appl. Phys. Lett. 86 124104

[9] Bermes C, Kim J-Y, Qu J and Jacobs L J 2007 Experimentalcharacterization of material nonlinearity using Lamb wavesAppl. Phys. Lett. 90 021901

[10] Bermes C, Kim J-Y, Qu J and Jacobs L J 2008 Nonlinear Lambwaves for the detection of material nonlinearity Mech. Syst.Signal Process. 22 638–46

[11] Deng M and Pei J 2007 Assessment of accumulated fatiguedamage in solid plates using nonlinear Lamb wave approachAppl. Phys. Lett. 90 121902

[12] Pruell C, Kim J-Y, Qu J and Jacobs L J 2007 Evaluation ofplasticity driven material damage using Lamb waves Appl.Phys. Lett. 91 231911

[13] Pruell C, Kim J-Y, Qu J and Jacobs L J 2008 A nonlinearguided wave technique for evaluating plasticity drivenmaterial damage in a metal plate NDT&E Int. at press

[14] Babin A and Figotin A 2001 nonlinear photonic crystals: I.Quadratic nonlinearity Waves Random Media 11 R31–102

[15] Boyd R 2003 Nonlinear Optics 2nd edn(New York: Academic)

[16] Deng M 1999 Cumulative second-harmonic generation ofLamb-mode propagation in a solid plate J. Appl. Phys.85 3051–8

[17] Deng M 2003 Analysis of second-harmonic generation ofLamb modes using a modal analysis approach J. Appl. Phys.94 4152–9

[18] Zhang T R, Choo H R and Dower C 1990 Phase and groupvelocity matching for second harmonic generation offemtosecond pulses Appl. Opt. 27 3927–33

[19] Imeshev G, Arbore M A, Fejer M M, Galvanauskas A,Fermann M and Harter D 2000 Ultrashort-pulse secondharmonic generation with longitudinally nonuniformquasi-phase matching gratings: pulse compression andshaping J. Opt. Soc. Am. B 17 304–18

[20] Fujioka N, Ashihara S, Ono H, Shimura T and Kuroda K 2005Group-velocity-matched noncollinear second-harmonicgeneration in quasi-phase matching J. Opt. Soc. Am. B22 1283–9

[21] Niethammer M, Jacobs L J, Qu J and Jarzynski J 2001Time-frequency representations of Lamb waves J. Acoust.Soc. Am. 109 1841–7

[22] Martin J F, Topper T H and Sinclair G M 1971 Computer basedsimulation of cyclic stress–strain behavior with applicationsto fatigue Mater. Res. Stand. 11 23–50

[23] Kim J-Y, Qu J, Jacobs L J, Littles J W and Savage M F 2006Acoustic nonlinearity parameter due to microplasticityJ. Nondestruct. Eval. 25 29–37

[24] Cantrell J H 2004 Substructural organization, dislocationplasticity and harmonic generation in cyclically stressedwavy slip metals Proc. R. Soc. A 460 757–80

7