Quantitative fatigue crack evaluation inpipeline structures using nonlinearcylindrical waves

Ruiqi Guan1, Ye Lu1,3 , Kai Wang2 and Zhongqing Su2

1Department of Civil Engineering, Monash University, Clayton, VIC 3800, Australia2Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, People’sRepublic of China

E-mail: ye.lu@monash.edu

Received 12 November 2018Accepted for publication 20 November 2018Published 20 December 2018

AbstractThis study comprehensively investigated the interaction between fatigue crack propagation andcylindrical guided waves which were excited and collected by surface-mounted piezoelectricelements on an aluminium pipe to identify the wave nonlinearity induced by the fatigue crack.Numerical simulation and experimental testing were conducted to quantify the relation betweenthe crack length and the wave nonlinearity collected in both transmission and reflectionconfigurations using an appropriate nonlinear index, where the co-existence of flexural wavemodes as a wave group at second harmonic was considered. Both numerical and experimentalresults revealed that the nonlinear index first increased monotonously and then it started to dropwith the further growth of the crack, a phenomenon which can be utilised for quantitativeassessment of initiation of fatigue cracks. Finally, the differences between the results fromsimulation and experiment were compared and discussed, where possible reasons for the changein nonlinear index in both approaches were elaborated.

Keywords: fatigue crack, nonlinear guided waves, pipeline structures

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

1. Introduction

Pipelines are crucial infrastructure which suffer from varioussorts of damage such as impact, corrosion and aging.Cylindrical guided waves are efficient for pipeline inspection,with the properties of long propagation range, strongrobustness and high sensitivity [1]. Research work has beenconducted to detect different types of gross damage [2–5] inpipes by linear wave characteristics. However, micro-scaledamage such as initial fatigue crack, corrosion at early stage,and material degradation cannot be detected in some cir-cumstances using linear wave based methods, restricted bythe wavelength of the excited wave mode. On the other hand,guided waves based on nonlinear characteristics are promis-ing in these cases, where nonlinear distortion caused by the

discontinuities, such as higher harmonic generation, sub-harmonic generation, and mixed frequency responses, can bedetected in the structure.

Applications of numerical simulation and experimentalobservation in terms of the nonlinearity from material andcontact acoustic nonlinearity (CAN) caused by damage inmetallic structures have shown great progression. Amongdifferent modelling approaches for wave and microcrackinteraction [6], a ’breathing’ crack model has been widelyutilised to explain the behaviour of a fatigue crack distortedby guided waves. To model material nonlinearity or CAN, thefinite element (FE) method has generally been used toinvestigate the relation between nonlinear parameters andmicrocrack length [7] and to locate microcracks [8] in platestructures. Studies from [7] with S0 mode excitation haveshown that the nonlinear parameter increases monotonouslywith crack length. Time reversal method using sensing data

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from FE modelling has been proposed to construct close crackimaging in an aluminium plate [9]. Another simulationmethod known as local interaction simulation approach(LISA) provided an efficient and optimised model for theclapping phenomenon of the fatigue crack when the waveinteracted with it in the plate [10]. Furthermore, LISA andCellular Automata for Elastodynamics (CAFE) modellingwere also applied in nonlinear media for analysing the wavepropagation and the differences between these two methodswere compared and discussed [11].

Further experimental testing has revealed the increasingrelation between the nonlinear parameter and the severity offatigue cracks through the mixed frequency response method[12] or the second harmonic generation method [13] in alu-minium specimens. Apart from metallic structures, the non-linearity detected by higher harmonic generation method wasalso studied in composites for delamination [14], thermalfatigue [15] and fatigue damage [16].

So far, however, studies conducted in pipes have beenmost focused on material nonlinearity. Analyses throughtheoretical and numerical models with the consideration ofhigher harmonic generation have been investigated compre-hensively, from arbitrary cross-sections nonlinear waveguides[17] to weakly nonlinear cylinders [18–20]. Simulation resultshave illustrated that the prerequisites for cumulative secondharmonic generation with different mode excitation in pipestructures were synchronism and non-zero power flux.Through experimental studies, an aluminium pipe underthermal fatigue damage was evaluated by the second har-monic generation method and the accumulation of secondharmonic was also observed with the increase in propagationdistance [21]. Furthermore, studies using torsional waves todetect cubic material nonlinearity in a pipe have shown lin-early increasing cumulative harmonic as the wave propaga-tion distance increased [22]. Apart from using waves thatpropagate axially in cylinders, experiments with nonlinearcircumferential guided waves have also been carried out tovalidate the severity evaluation of accumulated damage intubes using the second harmonic generation method [23].

However, limited studies using CAN have been under-taken for pipe-like structures, while the relation between theseverity of microcracks and the nonlinear parameter is still notwell established. In particular, the existence of multipleflexural wave modes in pipe-like structures further compli-cates the interaction between ’breathing’ crack and waves, incomparison with plate-like structures [24]. The main purposeof this study is therefore to quantify the relation between thenonlinear parameter and crack length in pipe structures usingnumerical and experimental analyses.

In this paper appropriate mode selection for the genera-tion of second harmonic in pipe structures is first introducedto provide a basis for the subsequent numerical and exper-imental studies. Numerical modelling and results are thenpresented, followed by a fatigue test as experimental ver-ification for comparison with the simulation results and forinvestigating the nonlinearity induced by a fatigue crack. Thecomparisons between simulation and experiment results areconducted and the possible reasons of differences are justified

by a further simulation study. Finally, the conclusions of thestudy are drawn.

2. Selection of excitation wave

For appropriate wave mode selection for the second harmonicgeneration, a dispersion curve of an aluminium pipe (30 mmouter diameter and 4 mm wall thickness) was plotted byDISPERSE, as in figure 1. Longitudinal wave mode L(0, 2)was selected as the fundamental wave due to its fastest groupvelocity and non-dispersive properties within a wide range offrequency. The excitation frequency was 300 kHz, such thatthe L(0, 2) mode would convert to the L(0, 3) mode at thedouble frequency 600 kHz. Since the velocities of those twowave modes were different, the cumulative effect caused bymaterial nonlinearity could be minimised and the secondharmonic generated by a fatigue crack could be highlighted. Itcould be noticed from the dispersion curve that a group offlexural wave modes F(n, 2) had group velocities similar tothat of the longitudinal wave mode L(0, 2), especially when n�2. Therefore, when a single surface-bonded actuator wasapplied, it would generate multiple modes at the excitationfrequency.

In this study, a notation for the classification of wavemodes from Rose [25] was adopted, where L(n, m) stands fora longitudinal mode group consisting of axisymmetric modesL(0, m) and non-axisymmetric modes F(n, m), where n isdefined as the circumferential order of a mode and m denotesthe group order. Therefore, the waves at fundamental fre-quency for the pipe under investigation were generated as agroup of wave modes L(n, 2) composed of a longitudinalmode L(0, 2) and flexural modes F(n, 2) (n=1, 2, 3K)which would then be converted by a fatigue crack to L(n, 3)modes at the second harmonic, consisting of L(0, 3) and F(n,3) modes.

3. Methods

3.1. 3D FE model

A 3D model in Abaqus/Explicit was utilised to simulate thewave propagation, material nonlinearity and the CAN in apipe. The pipe model dimensions were wall thickness 4 mm,outer diameter 30 mm and length 1 m. For fatigue crackinitiation, an 8 mm long 2 mm wide through-thickness notchwas modelled by removing corresponding elements from thepipe. A 6-cycle Hann-window tone burst signal with thecentral frequency at 300 kHz was excited and was applied aspoint loads at four edges of the actuator in the pipe axisdirection. A group of L(n, 2), including the longitudinal modeL(0, 2) and flexural modes F(n, 2) at the excited frequency,was generated due to the actuation of one transducer. Tworeceiving points were set separately at two sides of the notchso as to receive both wave transmission and reflection. Detailsof the locations of the transducers and monitoring points arepresented in figure 2.

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Figure 1. Dispersion curve of an aluminium pipe with 2030 mm diameter and 4 mm wall thickness. (a) Phase velocity; (b) Group velocity.

Figure 2. Arrangement of actuator and sensors (unit: mm).

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The element type in this model was selected as 3D eight-node brick elements (C3D8R) and with element size of 1 mm,which resulted in about 18 elements per wavelength of thefundamental wave mode L(0, 2) and 12 elements per wave-length of the second harmonic wave mode L(0, 3). On thebasis of the equation D /t L cgmax [26], a time step 5e-8 swas selected for accuracy and stability of the model, whereLmax is the maximum element size and cg is the fastest groupvelocity among all wave modes propagating in the model.

To simulate the ’breathing’ crack, built-in definitionknown as seam crack in Abaqus was set on the surfaces of the

Figure 3. Nonlinear wave simulation in finite element analysis (a) seam crack in simulation model; (b) waves pass through the seam crackwhen it closes; (c) waves are blocked by the seam crack when it opens.

Table 1. Material properties input for simulation model.

Material properties

Density (kg m−3) 2700Young’s mod-ulus (GPa)

68.9

Poisson’s ratio 0.33Third order elasticconstants (GPa)

A −320B −200C −190

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crack and then a surface-to-surface contact interaction wasassigned on the crack to achieve breathing behaviour sub-jected to propagating waves [8], as illustrated in figure 3(a),where the through-thickness fatigue crack exists at both tipsof the notch. In figure 3(b), it can be recognised that thecompressional part of the wave closes the crack and as a resultthe stress can pass through, whereas in figure 3(c), the ten-sional part of the wave opens the crack and the stress isblocked by the crack. In order to analyse the relation betweenthe nonlinear parameter and the crack length, the crack wasincreased from 1 to 14 mm in length at each tip with anincrement of 0.5, 1 or 2 mm in the circumferential directiontill it reached about half the circumference of the pipe. Sincethe material itself contains nonlinearity homogenously in thepipe in the experiment, apart from CAN from a ’breathing’crack, material nonlinearity was also included in the modelvia setting material properties with third order elastic con-stants [17], which were merged into the model through a usersubroutine file [8]. The basic material properties as well as thethird order elastic constants are listed in table 1.

3.2. Experimental procedure

An aluminium pipe with the same configurations as in thesimulation model was utilised in experiment. Here,5 mm×10 mm rectangular and 5 mm×5 mm square sur-face-bonded piezoelectric transducers were used as actuatorand receivers in the experiment, with the same arrangementsas in the simulation. A system for generating and collectingsignals was deployed in the experimental testing, whichincluded a Ritec RAM-5000 SNAP and an Agilent digitaloscilloscope. A high-power low-pass filter was connected tothe system so as to filter out harmonics higher than the fun-damental frequency from other sources.

With the same excitation signal as in simulation, thereflection and transmission wave signals were recorded beforethe fatigue test as benchmark signals. The pipe was thensubjected to the three-point bending on a fatigue machinewith a cyclic compressional load in the middle. The applied

load ranged from 0.2 to 2 kN, with the equivalent higheststress on the cross section at about 50% of the yield stress ofthe pipe, and the loading frequency was at 2 Hz. Duringfatigue testing, the signals were continuously collected every2000 cycles until 22 000 cycles, after which the interval forsignal collection was changed to 4000 cycles. Fatigue testingwas stopped at 42 000 cycles when the nonlinearity in thereceived signals dropped to noise level. Signals were recordedafter 1024 times average with a 200MHz sampling fre-quency. It should be noted that during each moment of signalcollection the fatigue machine was unloaded to avoid anyadverse effect of the applied loads on the opening and closurebehaviour of the fatigue crack.

4. Results

4.1. Transmission and reflection characteristics

4.1.1. Simulation results. A typical transmitted signal beforeintroduction of the simulated crack is in figure 4. It isobserved that more than one wave packet arrives in theinterest of time period because of the single actuatorexcitation. The first wave arrives at 83 μs, which is inaccordance with the group velocity of the L(0, 2) mode fromthe dispersion curve while the subsequent waves are flexuralmodes F(n, 2). To extract the waves at double frequency, aftercollecting all the signals, the input signal was flipped 180° (i.e. out-of-phase) and then was applied to the specimen againat the same location, followed by same signal collectionprocedures. These two received signals were added togetherto reduce the contribution of the fundamental wavecomponent and to highlight the second harmonic [27].Subsequently, each of the combined signals was processedwith a short-time Fourier transform (STFT) to extract theslices at double frequency which were then transformed backto the time domain by the inverse fast Fourier transform [8].Meanwhile, to obtain signals at the fundamental frequency,the received signals before combined with their out-of-phasecounterparts were directly processed with STFT and the slicesat fundamental frequency were also extracted. With such aprocedure, the time domain signals of interest at thefundamental and double frequencies were obtained forfurther analysis.

Transmitted signals after processing procedure as aboveare demonstrated in figure 5. In comparison with thebenchmark signal where only material nonlinearity wasintroduced, the cases with 1 and 2 mm seam cracks at thefundamental frequency show minor differences in figures 5(a)and (b), whereas the second harmonic wave arrives at the timelater than the fundamental one, mainly due to the reason thatthe group velocity of L(0, 3) induced by the microcrack atdouble frequency is slower than that of L(0, 2) at fundamentalfrequency. For both 1 and 2 mm seam crack cases, multiplewaves are induced at double frequency as in figure 5(c) beforethe arrival of L(0, 1), which is a group of wave modes L(n, 3),including the fastest L(0, 3) mode and F(n, 3) modes. It canbe calculated that the average amplitude level of the second

Figure 4. Typical transmitted signal from simulation.

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harmonic wave for the 2 mm crack case is about 1.4% of thewave at the fundamental frequency, slightly greater than thatof the 1 mm crack case, which is approximately 1.3% of thefundamental wave. However both are still much higher thanthat caused by material nonlinearity in benchmark as shownin figure 5(d). Similarly, nonlinearity induced by micorcrackwas also observed in the reflected signals at double frequency,as in figure 6. The average amplitude of the second harmonicwave of the 2 mm crack case is about 2.3% of the wave at thefundamental frequency and is greater than that of the 1 mmcrack case, and both of them are much higher than that in thebenchmark where material nonlinearity only is considered.

4.1.2. Experimental results. The transmitted signal beforefatigue test is shown in figure 7. The first wave packet arrivesalmost at the same time as that in simulation, and multiplewave packets known as L(n, 2) modes also appear before thearrival time of L(0, 1) mode at 126 μs, similar to theobservation in simulation results. The waveform of the signalreceived experimentally was somewhat different from the

simulation results, which however is still acceptable in theconsideration of inevitable difference between experimentand simulation such as the actual performance of thetransducer to efficiently generate different wave modes inexperiment.

As compared in figure 8(a) for the benchmark signalbefore the fatigue test and the signals at 12 000 and 22 000fatigue cycles, signals for the damage cases show no obviousdifference from the benchmark at fundamental frequency. Inthe comparison between those two damage cases at doublefrequency in figure 8(b), the average amplitude level of thesecond harmonic wave after 22 000 cycles is more than 2% ofthe wave at the fundamental frequency, which is obviouslyhigher than the case after 12 000 cycles; more importantly,both damage cases show greater nonlinearity than that in thebenchmark. Furthermore, it can be seen in figure 8(b) that thesecond harmonic wave from both damage cases arrives laterthan the benchmark signal and more than one wave packet isgenerated, which was similar with the simulation results.

Figure 5. Transmitted signals from simulation with CAN after STFT compared with benchmark signals (a) at fundamental frequency; (b)signals at fundamental frequency after zoomed in; (c) signals at double frequency and (d) benchmark signal at double frequency.

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On the other hand, the average amplitude of the secondharmonic wave of the benchmark, which is about 0.8% of thatat fundamental frequency, is much higher than that insimulation. This difference can be attributed to unavoidablenonlinearities from equipment and environment, together withthe material nonlinearity of the actual pipe which was notideally homogeneous. The reflected signals also show similarresults in figure 9, whereas the multimode second harmonicwaves are not as obvious as in the transmitted signals. This isbecause of the shorter distance between the sensor forreflection and the fatigue crack where the multiple wavemodes with close velocities are not completely separated.

4.2. Nonlinear index with growth of fatigue crack

4.2.1. Nonlinear parameter. From the analysis of signalsreceived in simulation and experiment, it could be observedthat the nonlinearity changed with the crack length and

Figure 6. Reflected signal from simulation with CAN after STFT compared with benchmark signals (a) at fundamental frequency; (b) atdouble frequency and (c) benchmark signal at double frequency.

Figure 7. Transmitted signal from experiment.

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fatigue cycle respectively. Generally, a relative nonlinearparameter β′=A2/A1

2 is used to assess the nonlinearity in asystem, where A1 and A2 are the amplitudes of fundamentalwave and second harmonic wave in the time domain [28, 29]or frequency domain [30, 31]. However, due to the multiplewave mode generation at fundamental and double frequenciesand the nonlinearity is mainly from the crack breathingbehaviour, the conventional relative nonlinear parameter isnot suitable to evaluate the CAN in pipe structures. A properparameter in the time domain was proposed in a previousstudy [24] to include all the wave packets induced by damagein order to evaluate quantitatively the nonlinearity induced bythe ’breathing’ crack in the pipe. In this study, the nonlinearparameter for both transmission and reflection was defined asthe integral of the amplitude profile of L(n, 3) before thearrival time of L(0, 1) at double frequency divided by theintegral of the amplitude profile of L(n, 2) before the arrival

time of L(0, 1) at fundamental frequency. Such a nonlinearindex was calculated for all the damage cases in experimentand simulation and was used as a measurement of thenonlinearity in the pipe so as to quantify the severity of thefatigue crack.

4.2.2. Nonlinear indices from simulation. As observed infigure 10, for both transmitted and reflected signals thenonlinear index increases monotonously with crack lengthuntil the crack length reaches around 6 mm at each notch tipwhere the index reaches a relatively constant value and then itbegins to drop to approximately half of the maximum value.The nonlinear index close to zero at 0 mm crack length is thebenchmark signal which contains material nonlinearity only,indicating that the material nonlinearity can be disregardedwhen compared with the damage cases. It is interesting thatthe total crack length for both tips (12 mm) when nonlinearity

Figure 8. Transmitted signals from experiment at different fatigue cycles compared with benchmark signal (a) at fundamental frequency; (b)at double frequency.

Figure 9. Reflected signals from experiment at different fatigue cycles compared with benchmark signal (a) at fundamental frequency; (b) atdouble frequency.

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reaches the maximum is close to the wavelength of the secondharmonic wave, which is around 13 mm. It was understoodthat the increase in nonlinearity was mainly caused by theincrease in the portion of the crack which could be completelyclosed and opened by the incident wave. However, when thecrack length became fairly long, e.g. longer than 6 mm in thisstudy, since the closure of the far edge of the crack dependson the waves that detour to it, vibration of the far edge of thecrack was minimised because the incident wave wasincapable of detouring to the middle part of the crack andonly the edge of the crack close to the actuator kept vibration.In such cases, part of the crack had evolved into an open

crack, where only the tip of the seam crack is still assumed toshow breathing behaviour which contributed to thenonlinearity. To confirm this assumption, signals atfundamental frequency were extracted for different crackcases as in figure 11. It can be seen that for models with 2 and6 mm crack, the amplitudes of fundamental wave were similarto the benchmark. However, the reflected and transmittedsignals start to change for the case of 8 mm crack, indicatingthe change of the close crack to the open one.

It is well-known that the second harmonic amplitudewould have a quadratic increase with input voltage level dueto material nonlinearity which can be described by a relative

Figure 10. Nonlinear index versus crack length at each tip of notch in simulation.

Figure 11. (a) Transmitted and (b) reflected signals at fundamental frequency from simulation model for benchmark, 2, 6 and 8 mm seamcrack cases.

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nonlinear parameter β′=A2/A12 [32]. To further figure out

the influence of input energy on the CAN, the numericalmodel was re-simulated with the input load ten times higherthan that in the original model. It is confirmed that the secondharmonic amplitude due to the CAN shows linear ratherquadratic increase with input voltage. Therefore, the trend ofnonlinear index in figure 10 will not change with variation ininput signals.

Two FE models based on the original one were furtherdeveloped to verify the reason of nonlinearity decrease infigure 10. The curves in figure 10 indicate that the nonlinearindices for 3 and 10 mm long cracks (at each tip of the notch)were similar. On the basis of the aforementioned assumption,the middle part (7 mm) of the 10 mm seam crack wouldbecome open when the waves passed through. Therefore, thelength of the notch was extended in the numerical model fromthe original 8 to 22 mm in the middle, where 14 mmrepresents the middle part of two 10 mm seam cracks(7 mm at each tip) that has become open. Thus only 3 mm

seam cracks at the two tips of the new notch were created, asshown in figure 12. In the updated model in figure 12(b), thepipe dimension, material properties and mesh setting were allthe same as in original model, as was the simulation methodof the ’breathing’ crack and material nonlinearity. Similarly,for the 4 and 8 mm crack cases which also show similarnonlinearity in figure 10, the same procedure was conductedfor the second updated model, where the length of the notchwas extended from the original 8 to 16 mm in the middle.

The resultant nonlinear index from the reflected signalsfor first updated model (for the 3 and 10 mm long crack cases)is 0.0228 (shown as a red dot in figure 10), which iscomparable to the nonlinear index from the 3 mm cracklength case at 0.0235 and the 10 mm crack length at 0.0224.Similar results are also observed from the transmitted signals(shown as a red square in figure 10), where the value from theupdated model is 0.0182, between 0.0171 (for 3 mm cracklength) and 0.0202 (for 10 mm crack length). Similarly,nonlinearity indices 0.0371 and 0.0279 for the reflected and

Figure 12. (a) Original model; (b) updated model with notch extended in the middle.

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transmitted signals were obtained in the second updatedmodel (for the 4 and 8 mm long crack cases), respectively(shown as a green dot and a green square in figure 10,respectively), which are also close to the values from themodel with 4 mm close crack. These results confirm theassumption that the decrease of nonlinearity in the simulationmodel was due to the disappearance of breathing behaviour inthe middle part of the crack, which becomes an open crack tothe incident guided waves.

4.2.3. Experimental results. The nonlinear indices were alsocalculated for all received signals in experiment underdifferent fatigue cycles as shown in figure 13, which iscurve-fitted using a fifth order polynomial function. It can beseen that the nonlinear index increases gradually with thefatigue cycle until about 26 000 cycles and then decreases tothe same level as in the benchmark. It is understood that theinitiation of fatigue crack includes the accumulation ofmaterial dislocation around the notch tips and the formationof microcracks, which contribute to the increase in wavenonlinearity. Then the crack developed from a microcrack toan open crack, no longer generating the CAN. It should be

mentioned that at the end of the fatigue test, the crack lengthobserved in the experiment was about 1 mm, which was onlyat the beginning stage of the simulation model. The increasingpart of nonlinear index with crack length from simulation andexperiment is both because of the increasing length of thecrack that contributes to the nonlinearity. However, thenonlinear index in the experiment began to drop at an earlystage, indicating that different to numerical simulation, thebreathing behaviour in practice existed only at the early stageof a fatigue crack, which would become an open crackimmediately upon growing to a macro scale.

To confirm this assumption, some typical transmitted andreflected signals at fundamental frequency are plotted infigure 14, including signals at benchmark, 12 000, 22 000,34 000 and 42 000 fatigue cycles. As evident in the figure, forthe transmitted L(0, 2) and F(n, 2) modes, the waveamplitudes are close to each other until 22 000 fatigue cyclesand the amplitudes at 34 000 and 42 000 fatigue cycles areobvious smaller than the other cases. Also, before 22 000fatigue cycles, there is no noticeable difference in thereflected L(0, 2) from the notch. However, the reflectedfundamental wave L(0, 2) clearly increases at 34 000 and42 000 cycles, revealing that the crack evolves from micro tomacro one and even the fundamental wave may be capable ofdetecting its existence.

5. Conclusions

The relation between cylindrical nonlinear waves and a fati-gue crack in an aluminium pipe was comprehensivelyinvestigated in this study with the consideration of the exis-tence of multiple wave modes in the pipe structure. Bothnumerical and experimental studies were carried out con-sidering material nonlinearity and CAN while the differencesbetween experimental and simulation results were discussed.In the simulation model, the seam crack length was con-tinuously increased to simulate growth of a fatigue crack. Forthe experimental testing while a fatigue machine was used forfatigue crack generation and piezoelectric transducers wereadopted for the acquisition of wave nonlinearity.

From the numerical simulation, it was found that thenonlinear index first increased monotonously with cracklength because of the increasing length of the ’breathing’crack but began to decrease at a specific crack length which isclose to the wavelength of the second harmonic wave. Afterthat the crack was evolved from micro to macro scale as anopen crack, where the signal changes in the fundamentalfrequency start to be dominant with the further growth of thecrack. However, in the experiment the nonlinear index startsto drop at early stage, indicating in practice the CAN onlyexists at initial stage of fatigue crack. The results from bothstudies show that the nonlinear index can quantify the non-linearity in a pipe structure caused by crack and it can detectthe initial stage of a fatigue crack within 1 mm in practice,providing higher sensitivity than linear guided waves formicrocrack detection.

Figure 13. Nonlinear index versus fatigue cycles from experimentaltesting for (a) transmitted signals; (b) reflected signals.

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Acknowledgments

Zhongqing Su is grateful to the Hong Kong Research GrantsCouncil for support via General Research Funds (No.15201416 and 15212417). This work was also partiallysupported by the National Natural Science Foundation ofChina (No. 51635008).

ORCID iDs

Ye Lu https://orcid.org/0000-0002-2319-7681Zhongqing Su https://orcid.org/0000-0002-0840-1891

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