researchonfatiguedamageinhigh-strengthsteel(fv520b...

Research ArticleResearch on Fatigue Damage in High-Strength Steel (FV520B)Using Nonlinear Ultrasonic Testing

Bingbing Chen Chao Wang Pengfei Wang Sanlong Zheng and Weiming Sun

Institute of Process Equipment and Control Engineering Zhejiang University of Technology Hangzhou 310023 China

Correspondence should be addressed to Pengfei Wang pfwangzjuteducn

Received 22 April 2020 Revised 15 July 2020 Accepted 3 August 2020 Published 19 August 2020

Academic Editor Yuri S Karinski

Copyright copy 2020 Bingbing Chen et alis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In view of the early fatigue damage of high-strength steel FV520B a nonlinear ultrasonic detection was performed on two types offatigue samples using nonlinear Lamb waves e experimental results indicated that the ultrasonic nonlinear parameter is highlysensitive to early fatigue damage in high-strength FV520B For plate specimens the ultrasonic nonlinear parameter increased withthe number of fatigue cycles Scanning electron microscopy (SEM) observations of the fatigue specimens revealed that as thenumber of fatigue cycles increased the microstructure of the material gradually deteriorated and the ultrasonic nonlinearparameter increased For notched specimens the ultrasonic nonlinear parameter increased as the size of the main crack increasedSEM observations of the fracture indicated that the ultrasonic nonlinear parameters were more consistent with the equivalentmicrocrack length (defined as the sum of microcrack lengths in the statistical area) as compared with the length of the main crackIt was determined that the nonlinear effect of the Lamb wave is related to the equivalent microcrack length inside the material andthat the ultrasonic nonlinear parameter can effectively characterize the fatigue damage state of high-strength FV520B

1 Introduction

High-strength FV520B is often used to make centrifugalcompressor impellers A centrifugal compressor is an im-portant energy conversion device and is widely used in thepetroleum chemical metallurgy natural gas transmissionaero-engine and mine ventilation industries and in otherimportant fields [1ndash3] e impeller is an integral part of thecentrifugal compressor associated with high rotationalspeeds and complex loading e impeller often generatessmall vibrations at high frequencies thus it is prone tofatigue failure [4 5] e accidents caused by fatigue failureare often catastrophic and can cause massive losseserefore the study of ultrasonic nondestructive testingtechnology for the early fatigue damage of high-strengthFV520B is of great significance

For most well-designed engineering components theearly damage of the material (before the formation ofmacrocracks) accounts for more than 80 of the com-ponent life [6] While the existing nondestructive testingtechnology which includes linear ultrasonic testing

technology is effective for the detection of macrodefects inmaterials it is less sensitive to early damage (eg dislo-cation structure and microcracks) to the materials beforemacrocracks appear in the fatigue process Recent studieshave shown that the degradation of the early mechanicalproperties of materials can trigger ultrasonic nonlineareffects [7] erefore the degradation can be used to detectthe damage degree of materials at an early stage of fatigueStudies have shown that the nonlinearity caused bymicrodefects (such as dislocations) is much greater thanthat caused by lattice anharmonicity Jhang and Kim [8]conducted nonlinear ultrasonic testing research on SS41and SS45 medium-carbon steel under tensile and fatigueloading respectively and analyzed the high-order har-monics in the process of ultrasonic propagation It wasfound that the nonlinear coefficient increased with in-creases in tensile load and fatigue cycle Deng and Pei [9]tested the fatigue damage of aluminum plates using anonlinear Lamb wave method and discovered that thestress wave factor decreased monotonously with in-creasing fatigue cycles Walker et al [10] used a nonlinear

HindawiShock and VibrationVolume 2020 Article ID 8847704 15 pageshttpsdoiorg10115520208847704

Rayleigh wave to characterize the damage to an A36 steelspecimen caused by a low-cycle tensile fatigue test e testresults showed that in the early stages of fatigue life thenonlinear coefficient increased with an increase of cyclenumber whereas when the fatigue cycle number washigher than 20 the nonlinear coefficient slowly decreasedZhang [11] tested (type) 304 austenitic stainless steelunder low-cycle fatigue using nonlinear ultrasound Itwas found that during the low-cycle fatigue damage of the304 stainless steel the ultrasonic nonlinear parametersshowed a trend wherein it first increased and then de-creased In the initial stages of fatigue damage the increasein the ultrasonic nonlinear parameters is mainly due to theincrease of dislocation density in the plane dislocationstructure and the subsequent decrease of the ultrasonicnonlinear parameters may be related to the formation ofdislocation walls and dislocation cell structures Wanget al [12] conducted nonlinear ultrasonic testing on threegroups of KMN high-strength steel fatigue specimens withdifferent cycles including a vibration fatigue test bendingfatigue test and tension bending fatigue test e resultsshowed that the nonlinear parameters initially increasedand then decreased with the increase of fatigue cycles emicrofracture characteristics of the KMN specimens wereanalyzed under different fatigue test types It was foundthat with the increase of fatigue cycle number the mi-crostructures of the KMN specimens gradually deterio-rated and cracks appeared Dutta et al [13] conductedultrasonic nonlinear detection on aluminum and steelspecimens a strong nonlinear effect from the crackedspecimen was observed and the magnitude of harmoniccomponents increased nonlinearly with the increasingamplitude of the input signal Shen et al [14ndash16] used thelocal interaction simulation approach (LISA) to study thecontact-impact clapping phenomena of the wave-crackinteractions based on the penalty method ey foundthat the higher harmonics were generated during thenonlinear interaction between fundamental waves andfatigue cracks e nonlinear scattering and mode con-version phenomenon of Lamb waves as they interact withfatigue crack were also studied Hong et al [17] establisheda model for simulating nonlinear characteristics of ul-trasonic waves propagating in a fatigued medium based onnonlinear constitutive relations of the medium and ver-ified it with experiments e simulation results consistedof the experimental results both of which show thatrelative acoustic nonlinearity parameter βprime linearly in-creased with the wave propagation distance due to thematerial and geometric nonlinearities Liu et al [18]simulated the models with different microcrack sizes andfound that Lyapunov exponent had a good linear rela-tionship with the size of the crack Wu et al [19 20]studied the nonlinear effect of the interaction of Lambwaves and cracks through experiments and simulationsResults showed that the relative nonlinear parameterlinearly increased with the crack propagation

e research indicated that nonlinear ultrasonic non-destructive testing technology can effectively detect earlydefects in a material and has broad application prospects

2 Theoretical Basis of NonlinearUltrasonic Testing

e example of a one-dimensional longitudinal wave isconsidered to demonstrate the formation of nonlinearultrasound in a solid For most types of materials therelationship between stress and strain is nonlinear [21]and can be described by Hookersquos law in small areas asfollows

σ Eε(1 + βε + middot middot middot) (1)

where E is the elastic modulus and β is the second-orderelastic coefficient (also called the nonlinear coefficient)

e equation for the motion of a particle in the solidmedium in the x-direction is as follows

ρz2u

zt2

zσzx

(2)

where ρ is the density of the medium x is the distance in thedirection of wave propagation t is the time and u is thedisplacement of the particle at location x in the mediumUsing (1) and (2) simultaneously and ignoring the higher-order terms that exceed the second-order term in (1) anequation for particle displacement can be obtained whichindicates the relationship between particle displacement andstrain as follows

z2u

zt2 c

2z2u

zx2 + 2c

2βzu

zx

z2u

zx2

(3)

where c is the wave velocity in the medium According to theperturbation theory we can assume that the displacement u

in (3) is determined by the following equation

u u1 + un1 (4)

where u1 is the solution when β 0 and un1 is the first-orderperturbation solution Equation (3) can now be written asfollows

z2u1

zt2 +

z2un1

zt2 c

2 z2u1

zx2 +

z2un1

zx21113888 1113889 1 + 2β

zu1

zx+

zun1

zx1113888 11138891113890 1113891

(5)

In this case the linear solution is set as follows

u1 A1 cos(kx minus ωt) (6)

where ω is the frequency and if λ is the wavelength thenk ωc 2πλ is the wave number e approximate ana-lytical solutions to (3) and (5) can then be obtained bycomprehensively applying the multiscale method and thetrial solution and neglecting the high-order small quantity inthe solution process as follows

u u1 + un1 A1 cos(kx minus ωt) minus A2 sin[2(kx minus ωt)]

(7)

where A1 and A2 represent the fundamental frequency andsecond harmonic amplitudes respectively and

2 Shock and Vibration

A2 β8A

21k

2x (8)

It can be observed from (7) that the second harmonicamplitude A2 describing the nonlinear response is related toβ which implies that β can be used as a parameter to describethe nonlinearity of the medium as shown in the followingequation

β 8

k2x

A2

A21

βpropA2

A21

(9)

erefore when the ultrasonic frequency and propa-gation distance are fixed the ultrasonic nonlinear param-eters can be calculated by measuring the amplitudes of thefundamental frequency and the second harmonic What wecare about is the change of nonlinear parameter For con-venience in this experiment the relative nonlinear pa-rameter βprime is used to replace the change of the ultrasonicnonlinear parameter

βprime A2

A21

(10)

In this article it is also called ultrasonic nonlinear pa-rameter β

3 Experimental Material andMeasurement Methods

31 Experimental Material FV520B steel is an importantmaterial for the manufacture of centrifugal compressorimpellers owing to its high strength high hardness goodwear resistance and other excellent mechanical properties Itis usually used in the manufacture of core parts in variouslarge machinery and equipment e main chemical com-positions andmechanical properties of FV520B are shown inTables 1 and 2

32 Fatigue Test Method e fatigue specimens weredesigned as plate specimens and notched specimens with athickness of 2mm e shapes of the plate and notch fatiguespecimens are shown in Figures 1 and 2 respectively esurface of each specimen was mechanically vibration-pol-ished with emery papers to keep the surface consistentbefore the fatigue test e tensile fatigue test was carried outon an electromagnetic resonance high-frequency fatiguetestingmachine to obtain a fatigue test sample with differentcycles e load waveform was a sine wave the stress ratiowas 01 and the frequency was 120Hz ere were nineplate-shaped specimens in total One was left as the originalspecimen and the rest were tested on the fatigue testingmachine e cycle times were 5times104 1times 105 5times1051times 106 5times106 7times106 1times 107 and 2times107 respectively andthe loading stress was 400MPa e finite element softwareANSYS was used to simulate the stresses on the notched

specimens the stress concentration factor at the notch was55e notched specimens were divided into two groups fortestinge two groups were loaded with stresses of 100MPaand 120MPa respectively and the stress concentrations atthe specimen notch were about 550MPa and 660MParespectively e simulation results are shown in Figure 3

33 Nonlinear Ultrasonic Testing In this experiment alongitudinal wave incidence method was used to excite anultrasonic Lamb wave which is highly sensitive to damageand ease of actuation e commonly used S1-S2 mode[22 23] was selected is mode is relatively easy to exciteand satisfies group velocity matching and has a high exci-tation efficiency It can be appropriately selected from theother Lamb wave modes as the group speed is high Anultrasonic Lamb wave measurement of the FV520B speci-mens was carried out using a RAM-5000 high-energy ul-trasonic system as shown in Figure 4 A schematic diagramof the nonlinear ultrasonic detection system is presented inFigure 5

e excitation signal of the nonlinear ultrasonic systemwas a 20-cycle Hamming-windowed sinusoidal tone-burstsignal e tone-burst signal was excited by a high-energyultrasound system and was then processed by an attenuatorand a low-pass filter before being transmitted to an ultra-sonic piezoelectric transducer e piezoelectric transducerconverted the voltage signal into an ultrasonic signal andtransmitted it to the specimen A piezoelectric transducer atthe receiving end converted the received ultrasonic signalinto a voltage signal e received signal was processed by ahigh-pass filter and a preamplifier and was sent to an os-cilloscope and computer for data processing and analysis Anarrow-band piezoelectric transducer with a center fre-quency of 225MHz and a wide-band piezoelectric trans-ducer with a center frequency of 5MHz were selected as thetransmitting and receiving probes respectively A 4MHzhigh-pass filter was used to filter the received signal A signalwith a frequency of 22MHz was used as the excitationsignal and the incident angle was 27deg e probe and thespecimen were coupled using glycerine amplitudes of thefundamental and second harmonic Lamb waves were ob-tained after the signals were processed by a short-timeFourier transform

4 Experimental Results

41 Dispersion Curve and Signal Validity Verificatione second harmonic generation efficiency is low owing tothe dispersion characteristics of the ultrasonic guided waveFurthermore the second harmonic signal was weak anddifficult to measure If the phase velocity of the fundamentalfrequency Lamb wave mode excited in the specimen wasequal to the phase velocity of the double-frequency Lambwave mode the second harmonic signal was relatively easyto measure erefore the group velocity and phase velocitydispersion curves of high-strength FV520B specimen wereobtained by solving the RayleighndashLamb dispersion equationusing MATLAB as shown in Figure 6 According to the

Shock and Vibration 3

Table 1 Chemical composition of FV520B (wt)

C Si Mn P S Ni Cr Cu Nb Mo Fele007 le007 le10 le003 le003 50ndash60 132ndash145 13ndash18 025ndash045 13ndash18 Ba1

Table 2 Mechanical properties of FV520B

Elasticity modulus E (GPa) Tensile strength Rm (MPa) Yield strength Rp02 (MPa) Vickers hardness HV (kgfmiddotmmminus2) Elongation δ ()194 1170 1029 380 1607

30

1005024472

10

R30

(a)30

10

035

1005024472

R012

153deg

04R30

(b)

Figure 1 Fatigue specimen size (a) Plate specimen (b) Notched specimen

(a) (b)

Figure 2 Photo of fatigue specimen (a) Plate specimen (b) Notched specimen

041406 12238 244718 367056 489395612106 183549 305887 428225 550564

(a)

00497 146893 293736 440579 587422734712 220314 367157 514 660844

(b)

Figure 3 e results of the finite element simulation at the specimen notch (a) 550MPa stress concentration (b) 660MPa stressconcentration


dispersion curve of the phase velocity the fundamentalfrequency of the Lamb wave matches the second harmonicphase velocity at 183MHz Due to the interference of ex-perimental equipment coupling agent and circuit thetheoretical excitation frequency will shift resulting in thedifference between theoretical calculation and actual

experimental measurement erefore frequency sweep ofhigh-strength FV520B specimen was carried out near thetheoretical frequency e amplitude of the second har-monic is the largest when the excitation frequency is22MHz so the excitation frequency of 22MHzwas selectedfor the nonlinear ultrasound experiment

(a) (b)

Figure 4 Nonlinear ultrasonic detection system (a) RAM-5000 high-energy ultrasonic system (b) Experiment procedure

Oscilloscope

Attenuator

LP filter

Amplifier

RAM SNAP 5000PC

HP filter

Plexiglas wedgeTx Rx

Figure 5 e schematic diagram of the nonlinear ultrasonic detection system

12

10

8

6

4

2

00 1 2 3 4 5

F (MHz)

A0

A1A2 A3

S0

S1 S2 S3

Phas

e velo

city

(km

s)

(a)

Gro

up v

eloci

ty (k

ms

)

5

4

3

2

1

00 1 2 3

F (MHz)4 5

A0 A1 A2 A3

S0 S1 S2 S3

(b)

Figure 6 Dispersion curve of the Lamb wave in the FV520B specimen (a) Phase velocity dispersion curve (b) Group velocity dispersioncurve


Before the nonlinear ultrasonic Lamb wave measure-ment of the FV520B specimens it was necessary to verify thesystem to ensure that the measured second harmonic signalwas caused by the test material rather than by the mea-surement system Nonlinear ultrasonic testing was carriedout on FV520B fatigue specimens and the time domainsignals were obtained as shown in Figure 7e time domainsignals were processed with STFT (short-time Fouriertransform) and the STFT time-frequency energy spectrumimage of FV520B specimen was obtained as shown inFigure 8 e STFT energy spectrum is represented by 256levels of gray scale and the deeper the color means thegreater the energy since the amplitude of the fundamentalwave and second harmonic wave can be obtained en thevalues of nonlinear parameter can be calculated e inci-dent voltage was kept constant and the distance between theincident transducer and the receiving transducer waschanged (from 40mm to 80mm) It was found that thenonlinear parameters of the ultrasound showed a linearlyincreasing trend with the increase of propagation distanceas shown in Figure 9 e results indicate that the secondharmonic signal received by the receiving transducer isgenerated by the fundamental frequency Lamb wavepropagation in the specimen rather than by the system orcouplant

42 Experimental Measurement Results e nonlinear pa-rameter β of the fatigue specimens with different fatiguecycles was divided by β0 (β0 is the ultrasonic nonlinearparameter of the original specimen) for normalization andthe normalized nonlinear parameter β was obtained enonlinear parameters βmentioned below are all normalizedvalues In order to reduce the error caused by nonlinear testthree times of nonlinear ultrasonic testing was carried outfor each fatigue specimen in the experiment and the averagevalue was taken as the test result e relationship betweenthe normalized nonlinear parameters and fatigue period isused to describe the nonlinear changes of materials owing tofatigue damage as shown in Figure 10 As seen inFigure 10(a) the nonlinearity parameter showed an in-creasing trend with an increase of fatigue cycles for the plate-shaped specimen (400MPa loading stress) For the fatiguetests of the notched specimens the relationship curve alsoshowed a similar trend Figure 10(b) shows the relationshipbetween the nonlinear parameters and the fatigue cycle ofthe notched specimen (550MPa stress concentration) Ascan be seen from Figure 10(b) the ultrasonic nonlinearparameters increased with the increase of fatigue cyclesHowever a significant decrease occurred at pointA Scanning electron microscopy (SEM) observation resultsindicated that the numbers and sizes of microcracks on thefracture surface of the specimen corresponding to point Awere significantly lower than those at other points Anexperiment with another group of notched specimens(660MPa stress concentration) also indicated that the ul-trasonic nonlinear parameters increase with the increase offatigue cycle as shown in Figure 10(c) e experimentalresults indicate that the ultrasonic nonlinear parameters are

highly sensitive to the fatigue damage of high-strengthFV520B steel e relationship between the ultrasonicnonlinear parameters and fatigue cycles can be used to

Excitation signal Received signal

Figure 7 Time domain signals of FV520B specimen (400MPaloading stress N 5times105)

5

4

3Fr

eque

ncy

(MH

z)

Time (s)

Fundamental wave

Second harmonic wave

00 20 times 10ndash5 40 times 10ndash5 60 times 10ndash5 80 times 10ndash5

2

1

0

Figure 8 STFT spectrograms of FV520B specimen Lamb wavesignals (400MPa loading stress N 5times105)

Nonlinearity parameter β

10

12

14

16

18

20

22

24

Non

linea

rity

para

met

er β

(Vndash1

)

50mm 60mm 70mm 80mm40mmPropagation distance (mm)

Figure 9 Relationship between nonlinear parameters and thepropagation distance


characterize the early fatigue degree of the material If theultrasonic nonlinear parameters of specific parts of thematerial are calibrated in advance it is expected that non-linear ultrasonic nondestructive testing technology can beused to detect the fatigue degree of in-service parts on aregular basis

5 Microstructure Observation and Discussion

51 Method and Sample for Microscopic Observation emain methods of microstructure observation include opticalmicroscopy and SEM For the plate specimens a crosssection of the specimen was observed using SEM especimen was cut in the middle position It was then inlaidpolished and ultrasonically cleaned before being observedby SEM e microscopic observation sample is shown inFigure 11 Zeiss field emission SEM was used to observe thesamples For the notched specimens the growth of the maincrack on the surface of the notch of the fatigue specimen was

observed under the optical microscope including the crackmorphology and the length of the main crack e notchedsample was then cut off using an Instron universal materialtesting machine and its section was observed under SEMe purpose of the microscopic observation is to comparethe microscopic damages of specimens with different fatiguedegrees erefore it is necessary to have the observationconditions as consistent as possible to ensure the accuracyof comparison e observation conditions include theobservation area and magnification

52 Microscopic Observations e specimens of high-strength FV520B with different fatigue cycles were observedby the aforementioned experimental instruments and mi-croscopic observation methods while maintaining the sameexperimental conditions as long as possible

Figure 12 shows the main crack morphology of thenotched specimen as observed under the metallographic

10121416182022242628

Nor

mal

ized

non

linea

r par

amet

er β

Normalized nonlinear parameter β

105 106 107104

Fatigue cycles N

(a)Normalized nonlinear parameter βA

2

4

6

8

10

12

14

16

Nor

mal

ized

non

linea

r par

amet

er β

10 times 105 15 times 105 20 times 105 25 times 105 30 times 10550 times 104

Fatigue cycles N

(b)


2

3

4

5

6

7

8

Nor

mal

ized

non

linea

r par

amet

er β

85 times 104 90 times 104 95 times 104 10 times 10580 times 104

Fatigue cycles N

(c)

Figure 10 Relationship between nonlinearity parameter and fatigue cycles (a) Relationship between nonlinearity parameter and fatiguecycles of plate specimen (400MPa loading stress) (b) Relationship between nonlinearity parameter and fatigue cycles of the notchedspecimen (550MPa stress concentration) (c) Relationship between nonlinearity parameter and fatigue cycles of the notched specimen(660MPa stress concentration)


microscope e propagation path of the main crack isperpendicular to the loading direction and the crack endsare bifurcated Figure 13 presents the fracture morphologyof the notched specimen e fracture surfaces of the fa-tigue crack propagation regions of high-strength FV520Bsteel are flat A large number of microcracks were found inthe crack source region and fatigue growth region and the

surface morphology of tensile fracture regions has adimpled shape e boundary between the crack propa-gation regions and the tensile fracture regions has an arcshape

e micrograph of the high-strength FV520B platespecimen (with 400MPa loading stress) under differentfatigue cycles is shown in Figure 14 As shown the

(a) (b)

Figure 11 Microscopic observation sample (a) Plate specimen (b) Notched specimen

Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)

Figure 12 Morphology of the main crack in notched specimen (a) Overall macroscopic appearance of the main crack (b) Crack near thenotch (c) Crack tip


Tensile fracture regions Crack propagationregions

(a)

Microcracks

(b)

Dimples

(c)

Figure 13 Fracture morphology of notched specimen (a) Overall appearance of cross section (b) Crack propagation regions (c) Tensilefracture regions

(a)

Pits

(b)

Figure 14 Continued


microstructure of the material deteriorates with the in-crease of fatigue cycles e matrix of the original specimenis relatively flat and has no evident defects and its cor-responding nonlinear parameters are relatively low Whenthe number of fatigue cycles reached 5times105 some smalldefects (such as pits) appeared in the material matrix andthe nonlinear parameters also increased With furtherincrease of fatigue cycles the number of microholes in-creased significantly microcracks began to appear and thenonlinear parameters continued to increase When thenumber of fatigue cycles increased to 2times107 the micro-cracks increased significantly and the nonlinear parame-ters continued to increase When the sinusoidal ultrasonicwave was transmitted into the solid medium a nonlinearinteraction occurred between the ultrasonic wave and thesolid medium leading to the generation of high-frequencyharmonics e generation of these harmonics is closelyrelated to the nonlinearity of the microstructure of the solidmedia and is usually caused by internal defects of materialssuch as dislocations micropores and cracks [24ndash26] Inthis experiment with the increase in the number of fatiguecycles the microstructures of the specimens graduallydeteriorated and the nonlinear parameters increased ac-cordingly erefore we can conclude that these deterio-rating microstructures (as evidenced by defects such asmicropores and cracks) lead to the generation of secondharmonics e results indicate that there is a certaincorrespondence between the nonlinear parameters and theinternal damage of the material and that the nonlinearparameters can characterize the fatigue damage of high-strength FV520B e micrographs of the FV520B notchedspecimens corresponding to 550MPa and 660MPa stressconcentrations under different fatigue cycles are shown inFigures 15 and 16 respectively As the specimen ultimatelyfractures the fracture is divided into a fatigue source zonefatigue crack expansion zone and last tensile fracture zonee variation of microcrack density can be obtained usingthe statistics of the microcracks It can be used to verify theexperimental results of the nonlinear ultrasound testingand to establish the relationship between the changes of the

ultrasonic nonlinear parameters and the changes of themicrostructure

53 Analysis of Microscopic Observations For the platespecimen (400MPa loading stress) with the increase of thenumber of fatigue cycles the microstructure of the specimengradually deteriorated as shown in Figure 14 For the ex-periments involving the two groups of notched specimensthe main crack propagated in the notch with an increase offatigue cycles as the specimen was in a state of stressconcentration in the notch e morphology and size of themain crack were measured using a metallographic micro-scope In addition with the increase of fatigue cycles thenumber and sizes of the microcracks increased eventuallyleading to the failure of the materials e process of in-creasing the fatigue cycle is associated with fatigue micro-crack initiation and propagation erefore we can observethe fracture surface of the specimen and calculate itsmicrocrack distribution

For this experiment it was necessary to count the crackdistributions of notched specimens with different fatiguecycles is process can be carried out in two steps First it isnecessary to select an appropriate and identical observationarea on the micrograph of each specimen As the crackdistributions in the micrographs of each specimen are notabsolutely uniform an area with clear cracks and uniformcrack distribution should be selected as the statistical area (asbest as possible) Second we calculate the number andlength of microcracks in the statistical region of the mi-crograph of each specimen As the sizes of themicrocracks inthe statistical area of the same micrograph can be differentwe should select clear and complete microcracks whencounting the number of microcracks Moreover as thestatistical area of each image is the same an equivalentmicrocrack length (ie the sum of microcrack lengths) canbe used to directly represent the changes in microcrackdensity e statistical results of the cracks in the notchedspecimens are shown in Tables 3 and 4 respectivelyFigure 17 shows the relationships between the main crack

Microcracks

(c)

Microcracks

(d)

Figure 14 Microstructure of plate specimen (400MPa loading stress) (a) Original microstructure (b) Microstructure of 5times105 cycles (c)Microstructure of 5times106 cycles (d) Microstructure of 2times107 cycles


Microcracks

(a)

Microcracks

(b)

Microcracks

(c)

Figure 15 Microstructure of notched specimens (550MPa stress concentration) (a) Microstructure of 100times105 cycles (b) Microstructureof 250times105 cycles (c) Microstructure of 275times105 cycles

Microcracks

(a)

Microcracks

(b)

Figure 16 Continued


Table 3 Statistics of the cracks in the notched specimens (550MPa stress concentration)

Samples number Sample no 1 Sample no 2 Sample no 3Main crack length (mm) 103 310 656Number of microcracks 25 10 29Length of largest microcrack (μm) 10993 7993 14883Equivalent microcrack length (μm) 142503 60268 18558



Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200

Main crack length L (mm)

Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β

Normalized nonlinear parameter βEquivalent crack length α

Figure 17 Relationship between equivalent microcrack length ultrasonic nonlinear parameters and main crack length of notchedspecimen (550MPa stress concentration)


length and the equivalent microcrack length and ultrasonicnonlinear parameters respectively

As shown in Figure 18 when the length of themain crackis less than 3mm the amplitude of the fundamental wavechanges slightly In contrast the ultrasonic nonlinear pa-rameters change significantly e equivalent microcracklength of the specimen cross section was calculated and itwas found that the equivalent microcrack length with theultrasonic nonlinear parameters had better consistency thanthe main crack length as shown in Figure 17 e ultrasonicnonlinear parameters increase with the increase of the lengthof the main crack but not monotonically When the lengthof the main crack reaches 31mm (corresponding to point Ain Figure 10(b)) the ultrasonic nonlinear parameters evi-dently decrease and the equivalent length of the microcrackalso shows corresponding changes is further indicatesthat the ultrasonic nonlinear effect is related to the

equivalent microcrack length in the specimen e ultra-sonic nonlinear parameters can well characterize thechanges of microcracks in high-strength FV520B and in-dicate the fatigue damage degree of the material Similarresults were obtained in the notched specimen (660MPastress concentration) experiment As shown in Figures 19and 20 with the increase of the main crack size the ul-trasonic nonlinear parameters were more sensitive than thefundamental amplitude e variation trends of the equiv-alent microcrack length and ultrasonic nonlinear parametershave better consistency

6 Conclusions

Nonlinear ultrasonic tests were performed on two types offatigue specimens (plate specimens and notched specimens)and the β-N curves of FV520B under three stress levels were

0 1 2 3 4 5 6 7ndash1Main crack length L (mm)

0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)

Normalized nonlinear parameter βAmplitude of fundamental wave

Figure 18 Ultrasonic nonlinear parameters of the notched specimen (550MPa stress concentration) with different main crack lengths

8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)

Normalized nonlinear paramater βAmplitude of fundamental wave

Figure 19 Ultrasonic nonlinear parameters of the notchedspecimen (660MPa stress concentration) with different main cracklengths

8

7

6

5

4

3

2

00 05 10 15 20 25 30 35 40Main crack length L (mm)

280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β

Normalized nonlinear parameter βEquivalent crack length α (μm)

Figure 20 Relationship between equivalent microcrack lengthultrasonic nonlinear parameters and main crack length of notchedspecimen (660MPa stress concentration)


obtained e results show that the ultrasound nonlinearparameter is highly sensitive to the early fatigue damage ofthe material

e microstructure was observed using SEMe resultsindicate that the change of ultrasonic nonlinear parametersis related to the deterioration of the microstructure of thematerial e nonlinear parameters can characterize thefatigue damage of FV520B material

e relationship between the ultrasonic nonlinear pa-rameters and the length of the main crack and equivalentmicrocrack length is analyzed As compared with the lengthof the main crack the equivalent microcrack length is moreconsistent with the ultrasonic nonlinear parameters indi-cating that the nonlinear parameters are mainly due to theappearance of the internal microcrack

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Authorsrsquo Contributions

BC conceptualized the study contributed to formal analysisand resources and was responsible for funding acquisitionCW contributed to methodology performed data curationand prepared the original draft CW and PW validated thestudy PW reviewed and edited the manuscript SZ per-formed study supervision WS was involved in projectadministration

Acknowledgments

is study was supported by the National Natural ScienceFoundation of China (no 51905484) (Research on very highcycle fatigue damage evaluation and life estimation methodof centrifugal compressor impeller based on nonlinear ul-trasonic testing) e paper was edited by Elsevier LanguageEditing Services

References

[1] W Q He ldquoFull-life mechanical response analysis of largecentrifugal compressor impellerrdquo Master thesis DalianUniversity Of Technology Dalian 2010

[2] L S Shu ldquoResearch on service life prediction model andnumerical simulation of centrifugal compressor remanufac-tured impellerrdquo Doctoral Dissertation Chongqing Univer-sity Chongqing China 2013

[3] M Zhang ldquoStudy on ultra high cycle fatigue behavior andmechanism of FV520B centrifugal compressor impeller ma-terialrdquo Doctoral dissertation Shandong University JinanChina 2015

[4] C W Wu Z Q Guan X L Guo et al ldquoFatigue reliabilityanalysis of large centrifugal compressor impeller bladesrdquoEquipment Manufacturing Technology vol 8 pp 1ndash3 2008

[5] Y Meng L Li and Q H Li ldquoTransient analysis method ofblade forced response under wake excitationrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 32pp 671ndash674 2006

[6] J H Cantrell ldquoSubstructural organization dislocation plas-ticity and harmonic generation in cyclically stressed wavy slipmetalsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 460no 2043 pp 757ndash780 2004

[7] G Shui J-Y Kim J Qu Y-S Wang and L J Jacobs ldquoA newtechnique for measuring the acoustic nonlinearity of materialsusing Rayleigh wavesrdquo NDT amp E International vol 41 no 5pp 326ndash329 2008

[8] K Jhang and K Kim ldquoEvaluation of material degradationusing nonlinear acoustic effectrdquo Ultrasonics vol 37 pp 39ndash44 1997

[9] M X Deng and J F Pei ldquoNonlinear ultrasonic Lamb waveresponse to fatigue of solid platesrdquo Acta Acoustics vol 33pp 360ndash369 2008

[10] S V Walker J Y Kim J Qu and L J Jacobs ldquoFatiguedamage evaluation in A36 steel using nonlinear Rayleighsurface wavesrdquo NDT amp E International Independent Non-destructive Testing and Evaluation vol 48 pp 10ndash15 2012

[11] J F Zhang ldquoStudy on nonlinear ultrasonic detection andevaluation of austenitic stainless steel service damagerdquoDoctoral Dissertation East China University of Science andTechnology Shanghai China 2014

[12] Z Wang P Qiao and B Shi ldquoNonpenetrating damageidentification using hybrid lamb wave modes from hilbert-huang spectrum in thin-walled structuresrdquo Shock and Vi-bration vol 2017 Article ID 5164594 11 pages 2017

[13] D Dutta H Sohn K A Harries and P Rizzo ldquoA nonlinearacoustic technique for crack detection in metallic structuresrdquoStructural Health Monitoring An International Journal vol 8no 3 pp 251ndash262 2009

[14] Y Shen J Wang and W Xu ldquoNonlinear features of guidedwave scattering from rivet hole nucleated fatigue cracksconsidering the rough contact surface conditionrdquo SmartMaterials and Structures vol 27 no 10 p 105044 2018

[15] Y Shen and C E S Cesnik ldquoNonlinear scattering and modeconversion of Lamb waves at breathing cracks an efficientnumerical approachrdquo Ultrasonics vol 94 pp 202ndash217 2019

[16] Y Shen and C E S Cesnik ldquoModeling of nonlinear interactionsbetween guided waves and fatigue cracks using local interactionsimulation approachrdquo Ultrasonics vol 74 pp 106ndash123 2017

[17] M Hong Z Su Q Wang L Cheng and X Qing ldquoModelingnonlinearities of ultrasonic waves for fatigue damage char-acterization theory simulation and experimental validationrdquoUltrasonics vol 54 no 3 pp 770ndash778 2014

[18] X Liu L Bo Y Liu et al ldquoDetection of micro-cracks usingnonlinear lamb waves based on the Duffing-Holmes systemrdquoJournal of Sound and Vibration vol 405 pp 175ndash186 2017

[19] Q Wu R Wang F Yu and Y Okabe ldquoApplication of anoptical fiber sensor for nonlinear ultrasonic evaluation offatigue crackrdquo IEEE Sensors Journal vol 19 no 13pp 4992ndash4999 2019

[20] R Wang Q Wu F Yu Y Okabe and K Xiong ldquoNonlinearultrasonic detection for evaluating fatigue crack in metal platerdquoStructural Health Monitoring vol 18 no 3 pp 869ndash881 2019

[21] K-Y Jhang ldquoNonlinear ultrasonic techniques for nonde-structive assessment of micro damage in material a reviewrdquoInternational Journal of Precision Engineering andManufacturing vol 10 no 1 pp 123ndash135 2009


[22] Y X Xiang M X Deng and F Z Xuan ldquoCreep damagecharacterization using nonlinear ultrasonic guided wavemethod a mesoscale modelrdquo Journal of Applied Physicsvol 115 p 044914 2014

[23] Y Xiang W Zhu C-J Liu F-Z Xuan Y-N Wang andW-C Kuang ldquoCreep degradation characterization of tita-nium alloy using nonlinear ultrasonic techniquerdquo NDT amp EInternational vol 72 pp 41ndash49 2015

[24] J Herrmann J-Y Kim L J Jacobs J Qu J W Littles andM F Savage ldquoAssessment of material damage in a nickel-basesuperalloy using nonlinear Rayleigh surface wavesrdquo Journal ofApplied Physics vol 99 no 12 p 124913 2006

[25] J-Y Kim L J Jacobs J Qu and J W Littles ldquoExperimentalcharacterization of fatigue damage in a nickel-base superalloyusing nonlinear ultrasonic wavesrdquo Ce Journal of theAcoustical Society of America vol 120 no 3 pp 1266ndash12732006

[26] W Li H Cui W Wen X Su and C C Engler-Pinto ldquoIn situnonlinear ultrasonic for very high cycle fatigue damagecharacterization of a cast aluminum alloyrdquo Materials Scienceand Engineering A vol 645 pp 248ndash254 2015


Rayleigh wave to characterize the damage to an A36 steelspecimen caused by a low-cycle tensile fatigue test e testresults showed that in the early stages of fatigue life thenonlinear coefficient increased with an increase of cyclenumber whereas when the fatigue cycle number washigher than 20 the nonlinear coefficient slowly decreasedZhang [11] tested (type) 304 austenitic stainless steelunder low-cycle fatigue using nonlinear ultrasound Itwas found that during the low-cycle fatigue damage of the304 stainless steel the ultrasonic nonlinear parametersshowed a trend wherein it first increased and then de-creased In the initial stages of fatigue damage the increasein the ultrasonic nonlinear parameters is mainly due to theincrease of dislocation density in the plane dislocationstructure and the subsequent decrease of the ultrasonicnonlinear parameters may be related to the formation ofdislocation walls and dislocation cell structures Wanget al [12] conducted nonlinear ultrasonic testing on threegroups of KMN high-strength steel fatigue specimens withdifferent cycles including a vibration fatigue test bendingfatigue test and tension bending fatigue test e resultsshowed that the nonlinear parameters initially increasedand then decreased with the increase of fatigue cycles emicrofracture characteristics of the KMN specimens wereanalyzed under different fatigue test types It was foundthat with the increase of fatigue cycle number the mi-crostructures of the KMN specimens gradually deterio-rated and cracks appeared Dutta et al [13] conductedultrasonic nonlinear detection on aluminum and steelspecimens a strong nonlinear effect from the crackedspecimen was observed and the magnitude of harmoniccomponents increased nonlinearly with the increasingamplitude of the input signal Shen et al [14ndash16] used thelocal interaction simulation approach (LISA) to study thecontact-impact clapping phenomena of the wave-crackinteractions based on the penalty method ey foundthat the higher harmonics were generated during thenonlinear interaction between fundamental waves andfatigue cracks e nonlinear scattering and mode con-version phenomenon of Lamb waves as they interact withfatigue crack were also studied Hong et al [17] establisheda model for simulating nonlinear characteristics of ul-trasonic waves propagating in a fatigued medium based onnonlinear constitutive relations of the medium and ver-ified it with experiments e simulation results consistedof the experimental results both of which show thatrelative acoustic nonlinearity parameter βprime linearly in-creased with the wave propagation distance due to thematerial and geometric nonlinearities Liu et al [18]simulated the models with different microcrack sizes andfound that Lyapunov exponent had a good linear rela-tionship with the size of the crack Wu et al [19 20]studied the nonlinear effect of the interaction of Lambwaves and cracks through experiments and simulationsResults showed that the relative nonlinear parameterlinearly increased with the crack propagation

e research indicated that nonlinear ultrasonic non-destructive testing technology can effectively detect earlydefects in a material and has broad application prospects

2 Theoretical Basis of NonlinearUltrasonic Testing

e example of a one-dimensional longitudinal wave isconsidered to demonstrate the formation of nonlinearultrasound in a solid For most types of materials therelationship between stress and strain is nonlinear [21]and can be described by Hookersquos law in small areas asfollows

σ Eε(1 + βε + middot middot middot) (1)

where E is the elastic modulus and β is the second-orderelastic coefficient (also called the nonlinear coefficient)

e equation for the motion of a particle in the solidmedium in the x-direction is as follows

ρz2u

zt2

zσzx

(2)

where ρ is the density of the medium x is the distance in thedirection of wave propagation t is the time and u is thedisplacement of the particle at location x in the mediumUsing (1) and (2) simultaneously and ignoring the higher-order terms that exceed the second-order term in (1) anequation for particle displacement can be obtained whichindicates the relationship between particle displacement andstrain as follows

z2u

zt2 c

2z2u

zx2 + 2c

2βzu

zx

z2u

zx2

(3)

where c is the wave velocity in the medium According to theperturbation theory we can assume that the displacement u

in (3) is determined by the following equation

u u1 + un1 (4)

where u1 is the solution when β 0 and un1 is the first-orderperturbation solution Equation (3) can now be written asfollows

z2u1

zt2 +

z2un1

zt2 c

2 z2u1

zx2 +

z2un1

zx21113888 1113889 1 + 2β

zu1

zx+

zun1

zx1113888 11138891113890 1113891

(5)

In this case the linear solution is set as follows

u1 A1 cos(kx minus ωt) (6)

where ω is the frequency and if λ is the wavelength thenk ωc 2πλ is the wave number e approximate ana-lytical solutions to (3) and (5) can then be obtained bycomprehensively applying the multiscale method and thetrial solution and neglecting the high-order small quantity inthe solution process as follows

u u1 + un1 A1 cos(kx minus ωt) minus A2 sin[2(kx minus ωt)]

(7)

where A1 and A2 represent the fundamental frequency andsecond harmonic amplitudes respectively and


A2 β8A

21k

2x (8)


β 8

k2x

A2

A21

βpropA2

A21

(9)


βprime A2

A21

(10)















30

1005024472

10

R30

(a)30

10

035

1005024472

R012

153deg

04R30

(b)


(a) (b)


041406 12238 244718 367056 489395612106 183549 305887 428225 550564

(a)

00497 146893 293736 440579 587422734712 220314 367157 514 660844

(b)





(a) (b)


Oscilloscope

Attenuator

LP filter

Amplifier

RAM SNAP 5000PC

HP filter



12

10

8

6

4

2

00 1 2 3 4 5

F (MHz)

A0

A1A2 A3

S0

S1 S2 S3

Phas

e velo

city

(km

s)

(a)

Gro

up v

eloci

ty (k

ms

)

5

4

3

2

1

00 1 2 3

F (MHz)4 5

A0 A1 A2 A3

S0 S1 S2 S3

(b)








5

4

3Fr

eque

ncy

(MH

z)

Time (s)

Fundamental wave



2

1

0



10

12

14

16

18

20

22

24

Non

linea

rity

para

met

er β

(Vndash1

)










10121416182022242628

Nor

mal

ized

non

linea

r par

amet

er β


105 106 107104

Fatigue cycles N


2

4

6

8

10

12

14

16

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(b)


2

3

4

5

6

7

8

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(c)






(a) (b)


Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)




(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References





























A2 β8A

21k

2x (8)


β 8

k2x

A2

A21

βpropA2

A21

(9)


βprime A2

A21

(10)















30

1005024472

10

R30

(a)30

10

035

1005024472

R012

153deg

04R30

(b)


(a) (b)


041406 12238 244718 367056 489395612106 183549 305887 428225 550564

(a)

00497 146893 293736 440579 587422734712 220314 367157 514 660844

(b)





(a) (b)


Oscilloscope

Attenuator

LP filter

Amplifier

RAM SNAP 5000PC

HP filter



12

10

8

6

4

2

00 1 2 3 4 5

F (MHz)

A0

A1A2 A3

S0

S1 S2 S3

Phas

e velo

city

(km

s)

(a)

Gro

up v

eloci

ty (k

ms

)

5

4

3

2

1

00 1 2 3

F (MHz)4 5

A0 A1 A2 A3

S0 S1 S2 S3

(b)








5

4

3Fr

eque

ncy

(MH

z)

Time (s)

Fundamental wave



2

1

0



10

12

14

16

18

20

22

24

Non

linea

rity

para

met

er β

(Vndash1

)










10121416182022242628

Nor

mal

ized

non

linea

r par

amet

er β


105 106 107104

Fatigue cycles N


2

4

6

8

10

12

14

16

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(b)


2

3

4

5

6

7

8

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(c)






(a) (b)


Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)




(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References

































30

1005024472

10

R30

(a)30

10

035

1005024472

R012

153deg

04R30

(b)


(a) (b)


041406 12238 244718 367056 489395612106 183549 305887 428225 550564

(a)

00497 146893 293736 440579 587422734712 220314 367157 514 660844

(b)





(a) (b)


Oscilloscope

Attenuator

LP filter

Amplifier

RAM SNAP 5000PC

HP filter



12

10

8

6

4

2

00 1 2 3 4 5

F (MHz)

A0

A1A2 A3

S0

S1 S2 S3

Phas

e velo

city

(km

s)

(a)

Gro

up v

eloci

ty (k

ms

)

5

4

3

2

1

00 1 2 3

F (MHz)4 5

A0 A1 A2 A3

S0 S1 S2 S3

(b)








5

4

3Fr

eque

ncy

(MH

z)

Time (s)

Fundamental wave



2

1

0



10

12

14

16

18

20

22

24

Non

linea

rity

para

met

er β

(Vndash1

)










10121416182022242628

Nor

mal

ized

non

linea

r par

amet

er β


105 106 107104

Fatigue cycles N


2

4

6

8

10

12

14

16

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(b)


2

3

4

5

6

7

8

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(c)






(a) (b)


Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)




(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References































(a) (b)


Oscilloscope

Attenuator

LP filter

Amplifier

RAM SNAP 5000PC

HP filter



12

10

8

6

4

2

00 1 2 3 4 5

F (MHz)

A0

A1A2 A3

S0

S1 S2 S3

Phas

e velo

city

(km

s)

(a)

Gro

up v

eloci

ty (k

ms

)

5

4

3

2

1

00 1 2 3

F (MHz)4 5

A0 A1 A2 A3

S0 S1 S2 S3

(b)








5

4

3Fr

eque

ncy

(MH

z)

Time (s)

Fundamental wave



2

1

0



10

12

14

16

18

20

22

24

Non

linea

rity

para

met

er β

(Vndash1

)










10121416182022242628

Nor

mal

ized

non

linea

r par

amet

er β


105 106 107104

Fatigue cycles N


2

4

6

8

10

12

14

16

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(b)


2

3

4

5

6

7

8

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(c)






(a) (b)


Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)




(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References


































5

4

3Fr

eque

ncy

(MH

z)

Time (s)

Fundamental wave



2

1

0



10

12

14

16

18

20

22

24

Non

linea

rity

para

met

er β

(Vndash1

)










10121416182022242628

Nor

mal

ized

non

linea

r par

amet

er β


105 106 107104

Fatigue cycles N


2

4

6

8

10

12

14

16

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(b)


2

3

4

5

6

7

8

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(c)






(a) (b)


Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)




(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References



































10121416182022242628

Nor

mal

ized

non

linea

r par

amet

er β


105 106 107104

Fatigue cycles N


2

4

6

8

10

12

14

16

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(b)


2

3

4

5

6

7

8

Nor

mal

ized

non

linea

r par

amet

er β


Fatigue cycles N

(c)






(a) (b)


Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)




(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References
































(a) (b)


Stress direction

1mm

(a)

005mm

(b)

005mm

Bifurcate

(c)




(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References






























(a)

Microcracks

(b)

Dimples

(c)


(a)

Pits

(b)

Figure 14 Continued






Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References

































Microcracks

(c)

Microcracks

(d)



Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References





























Microcracks

(a)

Microcracks

(b)

Microcracks

(c)


Microcracks

(a)

Microcracks

(b)

Figure 16 Continued






Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References

































Microcracks

(c)


16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7

60

80

100

120

140

160

180

200


Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References
































6 Conclusions



0

2

4

6

8

12

10

14

16

Nor

mal

ized

non

linea

rity

para

met

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2

1

0

ndash100 05 10 15 20 25 30 35 40


Nor

mal

ized

non

linea

r par

amat

er β

and

ampl

itude

of f

unda

men

tal w

ave (

Vndash1

)



8

7

6

5

4

3

2


280

240

200

160

120

80 Equi

vale

nt cr

ack

leng

th α

(μm

)

Nor

mal

ized

non

linea

r par

amet

er β







Data Availability






Acknowledgments


References
































Data Availability






Acknowledgments


References





























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