Characterization of Contact-Type Defects in Mortar Using a ...

Research ArticleCharacterization of Contact-Type Defects in Mortar Using aNonlinear Ultrasonic Method

Zhichao Nie 1 Kui Wang 12 Mingjie Zhao 12 and Xiao Sun 3

1Key Laboratory of Hydraulic and Waterway Engineering of Ministry of Education Chongqing Jiaotong UniversityChongqing 400074 China2Engineering Research Center of Diagnosis Technology and Instruments of Hydro-Construction Chongqing Jiaotong UniversityChongqing 400074 China3School of Hydraulic and Ecology Engineering Nanchang Institute of Technology Nanchang 330099 China

Correspondence should be addressed to Kui Wang anhuiwk163com

Received 7 September 2020 Revised 8 October 2020 Accepted 16 October 2020 Published 9 November 2020

Academic Editor Zhigang Zang

Copyright copy 2020 Zhichao Nie et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Second harmonic generation (SHG) is one of the common techniques in the nonlinear ultrasonic test e contact-type defectsplay an important role in material damage which are hard to be detected e traditional nonlinear parameter β used to evaluatethe micro damage in material is derived from the classical stress-strain relation which is more suitable for the anharmonicity ofcrystal rather than the contact-type defects Recently the theoretical model based on the bilinear stiffness law was derived and thevalidity and applicability need to be further studied For this purpose by the numerical method the contact interface in mortar ischaracterized based on the damage indicator c e relation between the excitation voltage and c is obtained Moreover the effectsof the crack length and orientation on the damage indicator c are also obtained e experimental method is also used tocharacterize the contact interface in mortar Combining with the existing work the results obtained in this article are discussedand further conclusions can be drawn e conclusions in this article provide potential of quantitative detection of the contactinterface and quality evaluation of bonding layers in materials

1 Introduction

e generation and development of the micro cracks lead tostress concentration playing an important role in materialdeterioration and failure [1] For decades nonlinear ultra-sonic methods have gained significant attention and theyoffer enormous potential for detecting and evaluating thechange in the microstructure of the materials [2ndash6] A lot ofstudies have proved that traditional linear ultrasonicmethods are infeasible for the micro damage detection in amaterial [6ndash8] However in this stage the mechanism andregularity of the contact acoustic nonlinear behavior causedby interface are still insufficient which should be studiedmore comprehensively

Experimental studies on the higher harmonic generation(HHG) technique have been extensively conducted [9 10]and rich results have been obtained for different damage

types (crystal dislocation precipitation micro cracks etc)and materials (metals rock concrete etc) [11ndash14] Initiallymost studies focused on the cubic nonlinearity which be-longs to the field of classical nonlinear ultrasound For twodecades it has been demonstrated that the nonlinearitygenerated by contact-type defects at the mesoscale can beorders of magnitude higher than that of the microscale[15 16]is provides the possibility of damage detection forthe contact interfaces In subsequent studies the HHGtechnique for contact interface detection has been exten-sively studied [17ndash19] and the nonlinear parameter β isgenerally used to characterize the damage However thereare still many barriers before applying this technique inpractical test For example there are a lot of theoreticalstudies on the SHG at the contact-type defects while thevalidation by the experiment is still insufficient In additionespecially for geomaterial and concrete the contact-type

HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 8832934 11 pageshttpsdoiorg10115520208832934

defects are extensively and randomly distributed while theunified baseline of the damage state of the materials is hardto determine [20ndash22] us the results cannot be comparedwith each other to obtain further quantitative conclusions[23ndash25]

eoretically three crack-wave interaction modellingapproaches (ie bilinear stiffness hysteresis and roughsurface contact) are commonly used to explain the HHGphenomenon which have been comprehensively reviewedby Broda et al [26] e three models have their own ad-vantages and disadvantages e assumption of the bilinearstiffness is hard to be satisfied unless the level of excitationvoltage is high enough to drive the cyclic open-closed be-havior of the surfaces e rough surface contact is toocomplex when dealing with solids with distributed micro-cracks Hysteresis is a phenomenological model and thephysical structure of the crack is not taken into consider-ation Recently the scattering effect accompanied with thenonlinear ultrasonic phenomenon generating at the contactsurface has attracted much attention [27ndash30] Based on therough surface contact the nonlinear ultrasonic effectsgenerated at the interface is studied and the results havebeen verified through experiment [31] In addition a novelapproach is proposed to analyze the scattered field at a planeinterface based on the rough surface model e corre-sponding solution is obtained by decomposing the scatteredfield into two componentse novel approach is simpler forthe less unknowns [30] Furthermore the explicit analyticalsolution of the scattered field is derived improving theprevious theoretical studies obtained by numerical method[32] In addition Zhao et al derived the nonclassicalnonlinear parameter based on a bilinear stiffness [33 34]However the proposed model is still insufficient for lack ofexperimental validation Besides β0 is a global indicatorwhich provides no information on the local damage inmaterials

In actual test acoustic nonlinearity is commonly in-troduced by the electronic system as well as the couplingbetween the transductors and specimens us it is difficultto distinguish the material nonlinearity from the obtainednonlinearity in experiment In general the numericalmethod can be considered as an effective method for thenonlinear ultrasonic study [35ndash39] By finite element sim-ulations the motion of interface obeys the contact laws andthe nonlinear effects of ultrasonic waves can be obtained ByFEM analysis the nonlinear ultrasonic technique is bene-ficial in detecting a closed crack with a different orientation[40] A finite difference time-domain technique is proposedto study the wave scattering caused by cracks with inter-acting faces and the accuracy and stability of the scheme inone-dimensional (1D) and two-dimensional (2D) spaces areverified [41] However numerical simulations are difficult toconsider all factors having effects on the results usconsiderable difference usually exists between the numericalresults and the actual test and the experiment method isnecessary to be a supplement

e objective of this study is to detect the contact interfacein materials based on the bilinear stiffness assumption Forthis purpose the damage indicator derived from the bilinear

stiffness laws is verified numerically e effects of excitationvoltage on the indicator c are investigated Furthermore theeffects of the interface length and orientation on the secondharmonic generation are also studiede experimental studyis conducted in mortar specimens and the regularities arecoincident with those of numerical results

e paper is structured as follows In Section 2 thetheoretical background of the nonclassical damage indicatorc is briefly introduced ereafter the finite elementmodelling is described in Section 3 and the numerical re-sults are given and discussed In Section 4 the nonlinearultrasonic test is carried out in mortar specimens and theresults are presented and discussed Finally the conclusionsof the study are presented

2 SHG at Contact Surface

e explicit expression of SHG can be derived based on thebilinear stiffness assumption Considering a longitudinalplane wave propagation in an isotropic and a linear elasticsolid with Youngrsquos modulus E0 the corresponding waveequation is given by

zσzx

E0z2u

zx2

(1)

e corresponding bilinear stiffness expression can beexpressed as

E E0 1 minus H ε minus ε01113872 1113873ΔEE0

1113890 1113891 (2)

where ε0 is the initial static contact strain H(middot) is theHeaviside step function and coefficient ΔE is the stiffnessweakened by the crack interface It is assumed that εgt 0 forthe tensile strain ΔE is given by the following equation

ΔE E0 minusdσdε

1113888 1113889εgt0

(3)

Substituting the bilinear stiffness expression into the 1Dwave equation the inhomogeneous wave equation can beobtained by

E0 minus H (dudx) minus ε01113872 1113873 E0 minus (dσdε)( 1113857

ρz2u

zx2

z2u

zt2 (4)

Assuming that a sine wave passes through the crack at anormal direction the corresponding boundary-initial valueproblem is calculated using the following equation

c2L minusΔEρ

Hzu

zxminus ε01113888 11138891113890 1113891

z2u

zx2

z2u

zt2 tgt 0 xgt 0

u(0 t) A1 sinωt

u(x 0) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

By solving the above equation the amplitude of secondharmonic A2 can be expressed as

2 Advances in Materials Science and Engineering

A2 A1ωx

3πcL

(6)

where ω is the angular frequency x is the propagationdistance and cL is the longitudinal velocity

e nonclassical damage indicator c based on the bi-linear stiffness law can be written as

c A2

A1

ωx

3πcL

(7)

It should be noted that equations (6) and (7) areestablished on the assumption of ΔE E0 at the interfaceand the tensile part of the cyclic load cannot pass through thesurface in this case

In addition several conclusions can be drawn accordingto the expression of the damage indicator c which are asfollows

(1) e damage indicator c is linear with the wavepropagation distance x from the nonlinear sourcewhich is the same as the classical nonlinear pa-rameter β e expression of β is demonstrated in thefollowing equation

β A2

A21

k2x

8 (8)

where k is the wavenumber of the incident waves(2) e damage indicator c is defined as the ratio of the

second harmonic amplitude A2 over the funda-mental A1 and the validity and reliability should befurther examined

3 Finite Element Model

31 Model Description By defining the bilinear stiffness lawat the surface the CAN generated at the surface can be

observed In addition the regularities of the crack length aswell as the orientation on the damage indicator c can also beobtained e commercial FEM software ABAQUS is usedfor these purposes

Herein a homogeneous and isotropic solid of 4times 4mm2

in a 2D numerical simulation is considered A contact in-terface of length L0 is introduced at the center of the domainas shown in Figure 1 e cyclic load excited at the topboundary of the model is a 7-cycle Hann-windowed lon-gitudinal pulse with the frequency of 500 kHz e nodelocated at the middle of the bottom boundary is selected tocollect the y-direction displacements e material prop-erties are given in Table 1 e mesh of the model has a sizeof 003mmis ensures that the considered wavelength λ2f

is discretized by enough elements e element type is a 4-node bilinear plane strain quadrilateral reduced integrationhourglass control (CPE4R) e mesh quality has beenchecked and no poor mesh exists in the model e timeincrement is 16 ns which satisfies the Courant-Friedrichs-Lewy stability condition Δtle amin

ρE

1113968(amin is the smallest

element dimension) and Nyquist-Shannon sampling the-orem fs gt 2fm (fs is the sampling frequency and fm is themaximum frequency)e total time period is 3times10minus 5 s FastFourier transform (FFT) is performed to obtain the am-plitudes of the fundamental A1 and second harmonic A2 Toeliminate the nonlinearity coming from the large dis-placement and rotation of the elements the geometricalnonlinearity is set to be off and both the linear and thequadratic bulk viscosity parameters are zero erefore itcan be concluded that the detected nonlinearity is solelygenerated by the contact behavior of the interface surfaces

32 Contact Laws e bilinear stress-strain law is assumedto be concerned with the stiffness asymmetry at the cracksurfaces which can be stated as

σ E0 minus ΔE( 1113857ε εgt ε0

E0ε εle ε01113896 (9)

It is assumed that εgt 0 is positive for the tensile straine weakened elastic modulus ΔE is dependent on thedamage degree in material When the interface is wellbonded (ie no damage exists in material) ΔE equals 0 inthe cyclic compression-tensile load When the interface isruptured ΔE will increase to E0 and the tensile componentsof the cyclic load are unable to pass through the surface Inthis numerical model the contact laws applied at the crackcan be stated as

σ 0 εgt 0

E0ε εle ε01113896 (10)

e above contact laws are coincident with the theo-retical background e prestress σ0 is assumed to be zero inthe model In addition it should be noted that the secondharmonic generation depends on the asymmetrical normalstiffness For consistency the surface is considered flat andfrictionless and the vertical displacement is received andanalyzed In addition the contact type of the interface is

Incident waves

y

xContact surface

The node for record

2mm

4mm

Figure 1 FEM model configuration

Advances in Materials Science and Engineering 3

ldquohard contactrdquo is indicates that the surfaces cannot in-terpenetrate into each other and the crack surface allowsseparation after the contact

e time and frequency domain spectra of the intact andcracked models are shown in Figure 2 e amplitude of theexcitation is 80 nm in this case and the length of the crack is1mm e elastic waves are normally incident at the in-terface It can be concluded that for the cracked sample the+y direction displacement of the bottom element of thesurface is obviously small is indicates that the tensilestrain is hard to pass through the unbonded interface It canbe observed from the frequency spectrum that significantsecond harmonic appears at the surface of the crack com-pared with the intact sample As mentioned above the finiteelement model has no geometric and material nonlinearityand the acoustic nonlinearity should come from the inter-action between the surfaces of the crack e above resultsfurther prove that the contact laws are effective which can befurther applied in the numerical study

33 Numerical Results

331 Effects of the Excitation Levels on c As introduced inthe theoretical background the damage indicator c is de-fined as the ratio of the second harmonic amplitudeA2 to thefundamental A1 and the damage indicator c is a constantwith increase of the excitation levels e FEM analysis willbe conducted to verify the analytical expression

e amplitude of the incident waves varies from 10 to150 nm with a step of 10 nm In addition the crack has alength of 1mm and is normal to the incident waves edamage indicator c for various excitation levels is presentedin Figure 3 It can be noted that when the amplitude ofincident wave is weak corresponding c is small With theamplitude increasing c increases sharply en c remainsstable in case of large enough amplitude applied It is knownthat the CAN effects will occur once the excitation levelbecomes greater than the threshold e strong cyclic loadnear the surface will lead the nonlinear effects Consideringthat the trend of c is in accordance with the CANmodel thecorrectness and validity of c can be examined by the FEMmodel

332 Effects of the Interface Length on the Damage Indicatorc e length of contact interface is characterized based onthe damage indicator c e crack length L0 is arranged from04 to 1mm with an interval of 01mm e incident wavesare normally excited at the top surface of the model edamage indicator c is determined by the linear fitting of A1and A2 e amplitude of the incident waves varies from 60to 100 nm with a step of 10 nm e damage indicator c forvarious crack lengths is shown in Figure 4 As can be seen inTable 2 the coefficient of determination R2 of each fit line is

nearly 1 which indicates that the excitation level is inde-pendent of the damage indicator c In addition Figure 5shows that the crack length has a significant effect on thedamage indicator c A larger damage indicator c will beobtained with increase of the interface length A significantpositive correlation between the damage indicator c and thecrack length L0 can be obtained

It is worth mentioning that the crack length is alsocharacterized by other damage indicators based on thenonlinear ultrasonic method [34] as shown in Figure 6 Itcan be concluded that the nonlinear effects are strongerwhen the crack becomes longer us the FEM resultsprovide potential of quantitively characterizing the contactinterface in materials

333 Effects of the Interface Angle on the Damage Indicatorc To understand the regularity of the crack orientation onthe damage indicator c seven crack orientations (α 0deg 15deg30deg 45deg 60deg 75deg and 90deg) are considered e crack length is1mm in this section Besides the amplitude of the incidentwaves varies from 60 to 100 nm with a step of 10 nmSimilarly the damage indicator c is determined by linearfitting of A1 and A2 With increase of the excitation level thedistribution of indicator c can be observed in Figure 7Figure 8 shows the damage indicator c versus the crackorientation It can be concluded that the damage indicator c

decreases as the crack angle increases According to thebilinear stiffness assumption the second harmonic is gen-erated by the clapping effect of the surface us the sig-nificant contact nonlinearity should appear in case of thewaves being normally incident at the surface

In other studies the similar regularity of the crackorientation on the second harmonic generation has beenobtained [35] As shown in Figure 9 in the case of P wavenormalized A1 and A2 are affected by the crack angleFundamental A1 increases with the interface angle while theopposite trend can be observed on the second harmonic A2us the regularity of the damage indicator c decreases withincrease of the interface angle which is consistent with thenumerical results in this paper

4 Experimental Procedure

41 Sample Preparation e size of the mortar specimens is100mmtimes 100mmtimes 100mm ree groups of cubic mortarspecimens were prepared in this experiment Group Acontains 3 intact mortar specimens (I1simI3) which is used tostudy the relation between the excitation voltage and thedamage indicator c Group B contains 6 specimens (L1simL6)e length of the interface is 1sim6 cm with an interval of1 cm e interface angle is 0deg in this group Group Ccontains 7 specimens (A1simA7) e crack orientation is0sim90deg with a step of 15deg e interface length is 4 cm e

Table 1 Material properties of the FEM model

Material Density ρ (kgm3) Youngrsquos modulus E (GPa) Poissonrsquos ratio v Friction coefficient μMortar 2200 211 0167 0


introduction procedure of the contact interface will be givenlater emixture proportions and physical properties of themortar specimens are given in Table 3 e Portland cement(PO 325) was used in this experiment

e procedure for the contact interface generation is asfollows After pouring of mortar a thin steel sheet withmachine oil on the surface vertically penetrated into themortar mixture All the steel sheets are 12 cm in length and1mm in thicknesse insertion depth is 100mmewidthof the steel sheets is 1sim6 cm en the steel sheet was re-moved out of the specimen carefully and the specimenswere not moved before being demolded It should be notedthat the removal of the steel sheets was gradually conducted

70 80 90 10060Excitation levels (nm)

005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

04 mm 05 mm 06 mm 07 mm

08 mm 09 mm 10 mm

Figure 4 Damage indicator c for various crack lengths

Table 2 e coefficient of determination R2 for the linear fitting ofA1 and A2

Crack length (mm) 04 05 06 07 08 09 1Coefficient ofdetermination R2 0999 1 1 1 0995 0993 0998

Damage indicator γ

05 06 07 08 09 1004Crack length (mm)

00

01

02

03

04

05

Dam

age i

ndic

ator

γ


0 05 1 15 2 25 3

3210

ndash1ndash2ndash3

Disp

lace

men

t (m

m)

times 10ndash5

times 10ndash5Time (s)

Intact modelCracked model

(a)

times 1060 02 04 06 08 1 12 14 16 18 2

Frequency (Hz)


012

01

008

006

004

002Spec

tral

ampl

itude

(b)

Figure 2 (a) Time domain (b) Frequency domain spectra comparison of the intact and the cracked numerical models

20 40 60 80 100 120 140 1600Excitation levels (nm)

00

01

02

03

04

Dam

age i

ndic

ator

γ

Figure 3 Damage indicator c for various excitation levels


once sheets arrived at the bottom of the sample and thewhole process of interface introduction took about oneminute Consequently an artificial contact interface was leftin the specimen due to the fluidity of the mixture especimens were demolded after 24 h followed by curing in astandard chamber (95 relative humidity and 20degC) until 28days of age e mortar specimens are shown in Figure 10

To confirm that the artificial interface is contact thepulse velocity of each specimen is tested and the relatedresults are shown in Figure 11 It is concluded that the wavevelocity remains steady in all specimens which indicates thatthe contact interface is at micro scale and cannot be detectedby the traditional ultrasonic method Moreover the speci-men A7 was cut in two halves after the test as shown inFigure 12 e trace along the height of specimen can be

observed It is noted that the damage degree graduallyweakened along the height of specimen because of the self-weight of the mixture In conclusion the interface within thespecimen can be considered contact and the damage isessentially at micro scale

It should be pointed out that equation (8) is establishedon the assumption of the surface rupture while the interfacewithin the mortar can be considered the stiffness reductionand the corresponding analytical expression can be given as[42]

A2 ct minus cc( 1113857A1xω

3πcL

(11)

where ct and cc represent the tension and compressionasymmetry in the elastic modulus respectively e samedamage indicator can be obtained in equations (8) and (11)which is defined as the ratio of A2 to A1

2 4 6 8 100Crack length (mm)

00

01

02

03

04

05

DI1

Figure 6 Damage index DI1 versus the crack lengths [34]


005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

0deg15deg30deg45deg

60deg75deg90deg

Figure 7 Amplitude of fundamental A1 versus second harmonicA2 for various crack angles

0 15 30 45 60 75 90ndash15Crack angle (deg)

00

01

02

03

04

05

Dam

age i

ndic

ator

γ

Figure 8 Distribution of damage indicator c for various crackangles

A 1A

inc

A 2A

inc

Normolized A1Normolized A2

20 40 60 80 1000Crack angle (deg)

055

060

065

070

075

080

085

090

095

100

004

006

008

010

012

014

Figure 9 Distribution of fundamental A1 and second harmonic A2for various crack angles [35]


42 Experimental Setup e experimental setup for themeasurement of fundamental A1 and second harmonic A2 isshown in Figure 13 According to the previous study [43] formortar specimens strong attenuation will happen when thecenter frequency is above 500 kHz erefore a tone-burstsignal of 10 cycles at 200 kHz is generated by a functiongenerator (Rigol 1022U) e pulse repetition rate is 10msTwo contact-type transducers (PXR 50) are used at thetransmitting and receiving ends Figure 14 shows that PXR 50has a good response to fundamental A1 and second harmonicA2 considering that the sensitivity at f0 and 2f0 is above 60 dBe receiving time domain signal after modulating isrecorded by a digital oscilloscope (Tektronix MDO 3104) andthe sampling rate is 50mss e quality and repeatability of

Table 3 Details of mixture proportions and physical properties of the mortar specimens

Design compressive strength (MPa) Average density (kgm3) wc ()Unit quantity (kgm3)

Average P wave velocity (ms)Water Cement Sand

10 1984 60 336 560 1680 3305

Figure 10 Mortar specimens for SHG test in experiment

Wave velocityWave velocity

I2 I3 A1 A2 A3 A4 A5 A6 A7 L1 L2 L3 L4 L5 L6I1Specimen number

2500

3000

3500

Wav

e vel

ocity

(ms

)

Figure 11 Distribution of wave velocity in each specimen

Figure 12 Longitudinal section diagram of the specimen A7

Figure 13 Test system for signal generation and acquisition

150 200 250 300 350 400100Frequency (kHz)

0

10

20

30

40

50

60

70

Sens

itivi

ty (d

B)

Figure 14 Schematic diagram of frequency response of PXR 50transducer


the measurements are improved by averaging the signals with256 acquisitions In addition a thin layer of Vaseline was usedas couplant between the transducers and the specimen sur-face e surface of the specimens was wiped carefully toensure no sand or dust is attached on the test point especimen and transducers were tightly wrapped during thetest To ensure that the transmitter and the receiver arealigned the test points were marked before the fixation oftransducers e FFT transform was performed on the timespectrum to obtain fundamental A1 and second harmonic A2Figure 15 shows an example of the time and frequency spectrameasured from a mortar specimen with interface respec-tively Fundamental A1 and second harmonic A2 can beobserved from the frequency spectrum and the damage in-dicator c can be calculated

43 Experimental Results and Discussion

431 Effects of the Excitation Levels on c e results of theintact mortar specimens are shown in Figure 16e value ofc is independent of the excitation voltage In addition thedamage indicator c in specimens I1simI3 can be determined bythe slope of the fit line being 0050 0043 and 0052 re-spectively e damage indicator c in specimens I1simI3 canbe regarded as the intrinsic nonlinearity of themortar whichcan be used to distinguish the nonlinearity coming from thecontact interface in group B and group C

e value of c is independent of the excitation voltagewhich has been observed by analyzing the leaky Rayleigh wavepassing through the crack surface [44] As shown in Figure 17the ratio of second harmonicA2 to fundamentalA1 is almost aconstant when σ is 371 kPa which indicates that the damageindicator c is independent of the excitation voltage It shouldbe noted that when a large σ is applied a minor gap existsbetween the two blocks In view of the damage type being thesame in the two studies the results are consistent with eachother even if the materials used in the studies are differentBesides the mechanism of second harmonic generation is notexplained theoretically in Vergara et alrsquos study

432 Effects of the Interface Length on the Damage Indicatorc e distribution of damage indicator c in different in-terface lengths is shown in Figure 18 c0 and δ represent the

006

004

002

0

ndash002

ndash004

ndash006

Am

plitu

de (V

)

0 1 2 3 4 5 6 7Time (s)

Hanning window

times10ndash5

(a)

A2A2

0 1 2 3 4 5 6 7Frequency (kHz) times105

60

50

40

30

20

10

Spec

tral

ampl

itude

A1

A2

(b)

Figure 15 Example of the SHG test (a) Time-domain signal (b) Frequency-domain signal

100 150 200 250 300 35050Excitation voltage (V)

000

001

002

003

004

005

006

007

008

Dam

age i

ndic

ator

γ

I1I2I3

Figure 16 Distribution of the excitation voltage and the damageindicator c in specimens I1simI3

A 1A

2

σ = 371kPa

200 250 300 350 400 450150Voltage (V)

0000

0002

0004

0006

0008

0010

Figure 17 Damage indicator c versus the excitation voltage [44]


means and standard deviations of the damage indicator c inintact mortar specimens c0plusmn δ is used to distinguish whichdamage indicators of the mortar specimens with interfacebehave different from the intact specimens It can be ob-tained that the damage indicator c is obviously larger thanthose of intact specimens for the 2ndash6 cm interface length Forthe 1 cm interface although the damage indicator c is in theband it is still larger than the means of c in intact mortarspecimens us it can be concluded that the damage in-dicator c is effective to characterize the contact interface inmaterials On the other hand positive correlation betweenthe nonlinear parameter c and the interface length L can alsobe revealed which is coincident with the numerical results

433 Effects of the Interface Angle on the Damage Indicatorc Figure 19 shows the distribution of the damage indicatorc in the mortar specimens with different interface orien-tations e meanings of c0 and δ are the same as in theprevious section In general the damage indicator c issignificantly affected by the interface angle e decreasingtrend can be observed obviously with the increase of theinterface angle which is consistent with the numerical re-sults When the interface degree is close to 90deg the differencebetween the intact and the interface becomes smaller Inactual applications it is reasonable to conduct the test atdifferent direction of the material to avoid missing the in-terface paralleling the propagation direction of ultrasound

5 Conclusions

In this paper the SHG technique is studied based on thebilinear stiffness assumption e main conclusions drawnin this paper can be stated as follows

(1) By the numerical and experimental method theregularity of the excitation levels on the damageindicator c is attained Both numerical and experi-mental results show that c is a constant when theamplitude is large In addition c is effective incharacterizing the contact-type defects in mortar

(2) e regularity of the interface length on the indicatorc is also studied e damage indicator c becomeslarger when the elastic waves are normally incident atthe surface It should be noted that the value of c isalso affected by the attenuation at the surface and theabsorption and scattering effects should be consid-ered to obtain the quantitative results

(3) e damage indicator c is effective in characterizingthe interface orientation c decreases with the in-crease of the interface angle Moreover the resultsare discussed with the existing work and can befurther verified and analyzed

Further in-depth research should be carried out on theexperimental studies of the CAN generation in materialsespecially the validation for the existing theoretical resultsIn addition quantitative characterization should be furtherstudied which will be of benefit to applying this technique inpractical test

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by Chongqing Research Programof Basic Research and Frontier Technology Grant nocstc2017jcyjBX0066 Key Laboratory of Hydraulic and

2 3 4 5 61Crack length L (cm)

004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ

γ of mortar samples with different lengthsγ0 + δγ0 ndash δ

Figure 18 Damage indicator c in specimens L1simL3

0 15 30 45 60 75 90ndash15Interface angle

004

005

006

007

008

009

Dam

age i

ndic

ator

γ

γ of mortar samples with different orientationsγ0 + δγ0 ndash δ

Figure 19 Damage indicator c in specimens A1simA7


Waterway Engineering of the Ministry of Education Grantno SLK2017B05 Innovative Research Program ofChongqing Jiaotong University Grant no 2019B0101Jiangxi Youth Science Fund Project Grant no20171BAB216042 Advantage Technology Innovation Teamof Jiangxi Province (Grant no 20171BCB24012) and Scienceand Technology Projects of Department of Transportation ofJiangxi Province Grant no 2018Q0028

References

[1] A Klepka W Staszewski R Jenal M Szwedo J Iwaniec andT Uhl ldquoNonlinear acoustics for fatigue crack detec-tionmdashexperimental investigations of vibro-acoustic wavemodulationsrdquo Structural Health Monitoring An InternationalJournal vol 11 no 2 pp 197ndash211 2012

[2] Y Yang C-T Ng and A Kotousov ldquoInfluence of crackopening and incident wave angle on second harmonic gen-eration of Lamb wavesrdquo Smart Materials and Structuresvol 27 no 5 Article ID 055013 15 pages 2018

[3] H Yan C Xu D Xiao and H Cai ldquoProperties of GH4169superalloy characterized by nonlinear ultrasonic wavesrdquoAdvances in Materials Science and Engineering vol 2015Article ID 457384 9 pages 2015

[4] Z Su C Zhou M Hong L Cheng Q Wang and X QingldquoAcousto-ultrasonics-based fatigue damage characterizationlinear versus nonlinear signal featuresrdquo Mechanical Systemand Signal Processing vol 45 no 1 pp 1ndash10 2014

[5] C Yang and J Chen ldquoFully noncontact nonlinear ultrasoniccharacterization of thermal damage in concrete and corre-lation with microscopic evidence of material crackingrdquo Ce-ment Concrete Research vol 123 Article ID 105797 10 pages2019

[6] J Chen C Yang and Q Guo ldquoEvaluation of surface cracks ofbending concrete using a fully non-contact air-couplednonlinear ultrasonic techniquerdquo Materials and Structuresvol 51 Article ID 104 9 pages 2018

[7] M Zhao Z Nie K Wang P Liu and X Zhang ldquoNonlinearultrasonic test of concrete cubes with induced crackrdquo Ul-trasonics vol 97 pp 1ndash10 2019

[8] C Mondal A Mukhopadhyay and R Sarkar ldquoA study onprecipitation characteristics induced strength variation bynonlinear ultrasonic parameterrdquo Journal of Applied Physicsvol 108 Article ID 124910 7 pages 2010

[9] M F Muller J-Y Kim J Qu and L J Jacobs ldquoCharac-teristics of second harmonic generation of Lamb waves innonlinear elastic platesrdquo e Journal of the Acoustical Societyof America vol 127 no 4 pp 2141ndash2152 2010

[10] J Kim D-G Song and K-Y Jhang ldquoAbsolute measurementand relative measurement of ultrasonic nonlinear parame-tersrdquo Research in Nondestructive Evaluation vol 28 no 4pp 211ndash225 2017

[11] M A Breazeale and D O ompson ldquoFinite-amplitudeultrasonic waves in aluminumrdquo Applied Physics Letters vol 3no 5 pp 77-78 1963

[12] A Hikata B B Chick and C Elbaum ldquoEffect of dislocationson finite amplitude ultrasonic waves in aluminumrdquo AppliedPhysics Letters vol 3 no 11 pp 195ndash197 1963

[13] M Fukuda and K Imano ldquoSecond harmonic ultrasonic wavesdetection using a double-layered piezoelectric transducerrdquoJournal of the Acoustical Society of America vol 140 no 4Article ID 3326 13 pages 2016

[14] P B Nagy ldquoFatigue damage assessment by nonlinear ultra-sonic materials characterizationrdquoUltrasonics vol 36 no 1ndash5pp 375ndash381 2018

[15] L A Ostrovsky S N Gurbatov and J N DidenkulovldquoNonlinear acoustics in nizhni novgorod (A review)rdquoAcoustical Physics vol 51 no 2 pp 114ndash127 2005

[16] I Y Solodov N Krohn and G Busse ldquoCAN an example ofnon-classical acoustic nonlinearity in solidsrdquo Ultrasonicsvol 40 no 1ndash8 pp 621ndash625 2002

[17] N Kim T-H Lee K-Y Jhang and I-K Park ldquoNonlinearbehavior of ultrasonic wave at crackrdquo AIP Conference Pro-cessing vol 1211 no 1 pp 313ndash318 2010

[18] Y Yang C-T Ng and A Kotousov ldquoBolted joint integritymonitoring with second harmonic generated by guidedwavesrdquo Structural Health Monitoring vol 18 no 1pp 193ndash204 2019

[19] V E Nazarov and A M Sutin ldquoNonlinear elastic constants ofsolids with cracksrdquo e Journal of the Acoustical Society ofAmerica vol 102 no 6 pp 3349ndash3354 1997

[20] J Chen YWu and C Yang ldquoDamage assessment of concreteusing a non-contact nonlinear wave modulation techniquerdquoNDT amp E International vol 106 pp 1ndash9 2019

[21] P Antonaci C L E Bruno A S Gliozzi and M ScalerandildquoMonitoring evolution of compressive damage in concretewith linear and nonlinear ultrasonic methodsrdquo Cement andConcrete Research vol 40 no 7 pp 1106ndash1113 2010

[22] J C Ongpeng A W C Oreta S Hirose and K NakahataldquoNonlinear ultrasonic investigation of concrete with varyingaggregate size under uniaxial compression loading andunloadingrdquo Journal of Materials in Civil Engineering vol 29no 2 Article ID 04016210 7 pages 2017

[23] P Liu H Sohn and B Park ldquoBaseline-free damage visual-ization using noncontact laser nonlinear ultrasonics and statespace geometrical changesrdquo Smart Materials and Structuresvol 25 Article ID 065036 12 pages 2015

[24] A A Shah and Y Ribakov ldquoNon-linear ultrasonic evaluationof damaged concrete based on higher order harmonic gen-erationrdquo Materials amp Design vol 30 no 10 pp 4095ndash41022009

[25] T Ju J D Achenbach L J Jacobs M Guimaraes and J QuldquoUltrasonic nondestructive evaluation of alkalindashsilica reactiondamage in concrete prism samplesrdquoMaterials and Structuresvol 50 Article ID 60 13 pages 2017

[26] D Broda W J Staszewski A Martowicz T Uhl andV V Silberschmidt ldquoModelling of nonlinear crack-waveinteractions for damage detection based on ultrasoundmdashareviewrdquo Journal of Sound and Vibration vol 333 no 4pp 1097ndash1118 2014

[27] L R F Rose P Blanloeuil M Veidt and C H WangldquoAnalytical and numerical modelling of non-collinear wavemixing at a contact interfacerdquo Journal of Sound and Vibrationvol 468 no 3 Article ID 115078 22 pages 2020

[28] K Manktelow R K Narisetti M J Leamy and M RuzzeneldquoFinite-element based perturbation analysis of wave propa-gation in nonlinear periodic structuresrdquo Mechanical Systemand Signal Processing vol 39 no 1-2 pp 32ndash46 2013

[29] K Manktelow M J Leamy and M Ruzzene ldquoComparison ofasymptotic and transfer matrix approaches for evaluatingintensity-dependent dispersion in nonlinear photonic andphononic crystalsrdquo Wave Motion vol 50 no 3 pp 494ndash5082013

[30] P Blanloeuil L R F Rose M Veidt and C H WangldquoAnalytical and numerical modelling of wave scattering by a


linear and nonlinear contact interfacerdquo Journal of Sound andVibration vol 456 pp 431ndash453 2019

[31] T Nam T Lee C Kim K-Y Jhang and N Kim ldquoHarmonicgeneration of an obliquely incident ultrasonic wave in solid-solid contact interfacesrdquo Ultrasonics vol 52 no 6pp 778ndash783 2012

[32] C Pecorari ldquoNonlinear interaction of plane ultrasonic waveswith an interface between rough surfaces in contactrdquo eJournal of the Acoustical Society of America vol 113 no 6pp 3065ndash3072 2003

[33] Y Zhao Y Qiu L J Jacobs and J Qu ldquoA micromechanicsmodel for the acoustic nonlinearity parameter in solids withdistributed micro cracksrdquo AIP Conference Processingvol 1706 no 1 9 pages Article ID 060001 2016

[34] Y Zhao F Li P Cao et al ldquoGeneration mechanism ofnonlinear ultrasonic Lamb waves in thin plates with randomlydistributed micro-cracksrdquo Ultrasonics vol 79 pp 60ndash672017

[35] P Blanloeuil A Meziane and C Bacon ldquoNumerical study ofnonlinear interaction between a crack and elastic waves underan oblique incidencerdquo Wave Motion vol 51 no 3pp 425ndash437 2014

[36] P Blanloeuil A J Croxford and A Meziane ldquoNumerical andexperimental study of the nonlinear interaction between ashear wave and a frictional interfacerdquo e Journal of theAcoustical Society of America vol 135 no 4 pp 1709ndash17162014

[37] P Blanloeuil L R F Rose M Veidt and C H Wang ldquoTimereversal invariance for a nonlinear scatterer exhibiting contactacoustic nonlinearityrdquo Journal of Sound and Vibrationvol 417 no 17 pp 413ndash431 2018

[38] X Wan P W Tse G H Xu T F Tao and Q ZhangldquoAnalytical and numerical studies of approximate phase ve-locity matching based nonlinear S0 mode Lamb waves for thedetection of evenly distributed microstructural changesrdquoSmart Materials and Structures vol 25 Article ID 04502320 pages 2016

[39] Z Nie K Wang and M Zhao ldquoApplication of wavelet andEEMD joint denoising in nonlinear ultrasonic testing ofconcreterdquo vol 2018 Article ID 7872036 11 pages 2018

[40] P Blanloeuil A Meziane A N Norris and C BaconldquoAnalytical extension of finite element solution for computingthe nonlinear far field of ultrasonic waves scattered by a closedcrackrdquo Wave Motion vol 66 pp 132ndash146 2016

[41] K Kazushi and I Yasuaki ldquoA finite difference method forelastic wave scattering by a planar crack with contactingfacesrdquo Wave Motion vol 52 pp 120ndash137 2015

[42] X Sun H Liu Y Zhao J Qu M Deng and N Hu ldquoe zero-frequency component of bulk waves in solids with randomlydistributed micro-cracksrdquo Ultrasonics vol 107 Article ID106172 8 pages 2020

[43] K Kawashima R Omote T Ito H Fujita and T ShimaldquoNonlinear acoustic response through minute surface cracksFEM simulation and experimentationrdquo Ultrasonics vol 40no 1ndash8 pp 611ndash615 2002

[44] L Vergara R Miralles J Gosalbez et al ldquoNDE ultrasonicmethods to characterise the porosity of mortarrdquo NDT amp EInternational vol 34 no 8 pp 557ndash562 2001


defects are extensively and randomly distributed while theunified baseline of the damage state of the materials is hardto determine [20ndash22] us the results cannot be comparedwith each other to obtain further quantitative conclusions[23ndash25]

eoretically three crack-wave interaction modellingapproaches (ie bilinear stiffness hysteresis and roughsurface contact) are commonly used to explain the HHGphenomenon which have been comprehensively reviewedby Broda et al [26] e three models have their own ad-vantages and disadvantages e assumption of the bilinearstiffness is hard to be satisfied unless the level of excitationvoltage is high enough to drive the cyclic open-closed be-havior of the surfaces e rough surface contact is toocomplex when dealing with solids with distributed micro-cracks Hysteresis is a phenomenological model and thephysical structure of the crack is not taken into consider-ation Recently the scattering effect accompanied with thenonlinear ultrasonic phenomenon generating at the contactsurface has attracted much attention [27ndash30] Based on therough surface contact the nonlinear ultrasonic effectsgenerated at the interface is studied and the results havebeen verified through experiment [31] In addition a novelapproach is proposed to analyze the scattered field at a planeinterface based on the rough surface model e corre-sponding solution is obtained by decomposing the scatteredfield into two componentse novel approach is simpler forthe less unknowns [30] Furthermore the explicit analyticalsolution of the scattered field is derived improving theprevious theoretical studies obtained by numerical method[32] In addition Zhao et al derived the nonclassicalnonlinear parameter based on a bilinear stiffness [33 34]However the proposed model is still insufficient for lack ofexperimental validation Besides β0 is a global indicatorwhich provides no information on the local damage inmaterials

In actual test acoustic nonlinearity is commonly in-troduced by the electronic system as well as the couplingbetween the transductors and specimens us it is difficultto distinguish the material nonlinearity from the obtainednonlinearity in experiment In general the numericalmethod can be considered as an effective method for thenonlinear ultrasonic study [35ndash39] By finite element sim-ulations the motion of interface obeys the contact laws andthe nonlinear effects of ultrasonic waves can be obtained ByFEM analysis the nonlinear ultrasonic technique is bene-ficial in detecting a closed crack with a different orientation[40] A finite difference time-domain technique is proposedto study the wave scattering caused by cracks with inter-acting faces and the accuracy and stability of the scheme inone-dimensional (1D) and two-dimensional (2D) spaces areverified [41] However numerical simulations are difficult toconsider all factors having effects on the results usconsiderable difference usually exists between the numericalresults and the actual test and the experiment method isnecessary to be a supplement

e objective of this study is to detect the contact interfacein materials based on the bilinear stiffness assumption Forthis purpose the damage indicator derived from the bilinear

stiffness laws is verified numerically e effects of excitationvoltage on the indicator c are investigated Furthermore theeffects of the interface length and orientation on the secondharmonic generation are also studiede experimental studyis conducted in mortar specimens and the regularities arecoincident with those of numerical results

e paper is structured as follows In Section 2 thetheoretical background of the nonclassical damage indicatorc is briefly introduced ereafter the finite elementmodelling is described in Section 3 and the numerical re-sults are given and discussed In Section 4 the nonlinearultrasonic test is carried out in mortar specimens and theresults are presented and discussed Finally the conclusionsof the study are presented

2 SHG at Contact Surface

e explicit expression of SHG can be derived based on thebilinear stiffness assumption Considering a longitudinalplane wave propagation in an isotropic and a linear elasticsolid with Youngrsquos modulus E0 the corresponding waveequation is given by

zσzx

E0z2u

zx2

(1)

e corresponding bilinear stiffness expression can beexpressed as

E E0 1 minus H ε minus ε01113872 1113873ΔEE0

1113890 1113891 (2)

where ε0 is the initial static contact strain H(middot) is theHeaviside step function and coefficient ΔE is the stiffnessweakened by the crack interface It is assumed that εgt 0 forthe tensile strain ΔE is given by the following equation

ΔE E0 minusdσdε

1113888 1113889εgt0

(3)

Substituting the bilinear stiffness expression into the 1Dwave equation the inhomogeneous wave equation can beobtained by

E0 minus H (dudx) minus ε01113872 1113873 E0 minus (dσdε)( 1113857

ρz2u

zx2

z2u

zt2 (4)

Assuming that a sine wave passes through the crack at anormal direction the corresponding boundary-initial valueproblem is calculated using the following equation

c2L minusΔEρ

Hzu

zxminus ε01113888 11138891113890 1113891

z2u

zx2

z2u

zt2 tgt 0 xgt 0

u(0 t) A1 sinωt

u(x 0) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

By solving the above equation the amplitude of secondharmonic A2 can be expressed as


A2 A1ωx

3πcL

(6)



c A2

A1

ωx

3πcL

(7)




β A2

A21

k2x

8 (8)









ρE





E0ε εle ε01113896 (9)


σ 0 εgt 0

E0ε εle ε01113896 (10)


Incident waves

y

xContact surface

The node for record

2mm

4mm






















005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

04 mm 05 mm 06 mm 07 mm

08 mm 09 mm 10 mm




Damage indicator γ

05 06 07 08 09 1004Crack length (mm)

00

01

02

03

04

05

Dam

age i

ndic

ator

γ


0 05 1 15 2 25 3

3210

ndash1ndash2ndash3

Disp

lace

men

t (m

m)

times 10ndash5



(a)

times 1060 02 04 06 08 1 12 14 16 18 2

Frequency (Hz)


012

01

008

006

004

002Spec

tral

ampl

itude

(b)



00

01

02

03

04

Dam

age i

ndic

ator

γ








3πcL

(11)



00

01

02

03

04

05

DI1



005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

0deg15deg30deg45deg

60deg75deg90deg



00

01

02

03

04

05

Dam

age i

ndic

ator

γ


A 1A

inc

A 2A

inc


20 40 60 80 1000Crack angle (deg)

055

060

065

070

075

080

085

090

095

100

004

006

008

010

012

014







10 1984 60 336 560 1680 3305




2500

3000

3500

Wav

e vel

ocity

(ms

)




150 200 250 300 350 400100Frequency (kHz)

0

10

20

30

40

50

60

70

Sens

itivi

ty (d

B)








006

004

002

0

ndash002

ndash004

ndash006

Am

plitu

de (V

)

0 1 2 3 4 5 6 7Time (s)

Hanning window

times10ndash5

(a)

A2A2


60

50

40

30

20

10

Spec

tral

ampl

itude

A1

A2

(b)



000

001

002

003

004

005

006

007

008

Dam

age i

ndic

ator

γ

I1I2I3


A 1A

2

σ = 371kPa

200 250 300 350 400 450150Voltage (V)

0000

0002

0004

0006

0008

0010





5 Conclusions






Data Availability




Acknowledgments



004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ




004

005

006

007

008

009

Dam

age i

ndic

ator

γ





References
















































A2 A1ωx

3πcL

(6)



c A2

A1

ωx

3πcL

(7)




β A2

A21

k2x

8 (8)









ρE





E0ε εle ε01113896 (9)


σ 0 εgt 0

E0ε εle ε01113896 (10)


Incident waves

y

xContact surface

The node for record

2mm

4mm






















005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

04 mm 05 mm 06 mm 07 mm

08 mm 09 mm 10 mm




Damage indicator γ

05 06 07 08 09 1004Crack length (mm)

00

01

02

03

04

05

Dam

age i

ndic

ator

γ


0 05 1 15 2 25 3

3210

ndash1ndash2ndash3

Disp

lace

men

t (m

m)

times 10ndash5



(a)

times 1060 02 04 06 08 1 12 14 16 18 2

Frequency (Hz)


012

01

008

006

004

002Spec

tral

ampl

itude

(b)



00

01

02

03

04

Dam

age i

ndic

ator

γ








3πcL

(11)



00

01

02

03

04

05

DI1



005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

0deg15deg30deg45deg

60deg75deg90deg



00

01

02

03

04

05

Dam

age i

ndic

ator

γ


A 1A

inc

A 2A

inc


20 40 60 80 1000Crack angle (deg)

055

060

065

070

075

080

085

090

095

100

004

006

008

010

012

014







10 1984 60 336 560 1680 3305




2500

3000

3500

Wav

e vel

ocity

(ms

)




150 200 250 300 350 400100Frequency (kHz)

0

10

20

30

40

50

60

70

Sens

itivi

ty (d

B)








006

004

002

0

ndash002

ndash004

ndash006

Am

plitu

de (V

)

0 1 2 3 4 5 6 7Time (s)

Hanning window

times10ndash5

(a)

A2A2


60

50

40

30

20

10

Spec

tral

ampl

itude

A1

A2

(b)



000

001

002

003

004

005

006

007

008

Dam

age i

ndic

ator

γ

I1I2I3


A 1A

2

σ = 371kPa

200 250 300 350 400 450150Voltage (V)

0000

0002

0004

0006

0008

0010





5 Conclusions






Data Availability




Acknowledgments



004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ




004

005

006

007

008

009

Dam

age i

ndic

ator

γ





References



































































005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

04 mm 05 mm 06 mm 07 mm

08 mm 09 mm 10 mm




Damage indicator γ

05 06 07 08 09 1004Crack length (mm)

00

01

02

03

04

05

Dam

age i

ndic

ator

γ


0 05 1 15 2 25 3

3210

ndash1ndash2ndash3

Disp

lace

men

t (m

m)

times 10ndash5



(a)

times 1060 02 04 06 08 1 12 14 16 18 2

Frequency (Hz)


012

01

008

006

004

002Spec

tral

ampl

itude

(b)



00

01

02

03

04

Dam

age i

ndic

ator

γ








3πcL

(11)



00

01

02

03

04

05

DI1



005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

0deg15deg30deg45deg

60deg75deg90deg



00

01

02

03

04

05

Dam

age i

ndic

ator

γ


A 1A

inc

A 2A

inc


20 40 60 80 1000Crack angle (deg)

055

060

065

070

075

080

085

090

095

100

004

006

008

010

012

014







10 1984 60 336 560 1680 3305




2500

3000

3500

Wav

e vel

ocity

(ms

)




150 200 250 300 350 400100Frequency (kHz)

0

10

20

30

40

50

60

70

Sens

itivi

ty (d

B)








006

004

002

0

ndash002

ndash004

ndash006

Am

plitu

de (V

)

0 1 2 3 4 5 6 7Time (s)

Hanning window

times10ndash5

(a)

A2A2


60

50

40

30

20

10

Spec

tral

ampl

itude

A1

A2

(b)



000

001

002

003

004

005

006

007

008

Dam

age i

ndic

ator

γ

I1I2I3


A 1A

2

σ = 371kPa

200 250 300 350 400 450150Voltage (V)

0000

0002

0004

0006

0008

0010





5 Conclusions






Data Availability




Acknowledgments



004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ




004

005

006

007

008

009

Dam

age i

ndic

ator

γ





References





















































3πcL

(11)



00

01

02

03

04

05

DI1



005

010

015

020

025

030

035

040

Dam

age i

ndic

ator

γ

0deg15deg30deg45deg

60deg75deg90deg



00

01

02

03

04

05

Dam

age i

ndic

ator

γ


A 1A

inc

A 2A

inc


20 40 60 80 1000Crack angle (deg)

055

060

065

070

075

080

085

090

095

100

004

006

008

010

012

014







10 1984 60 336 560 1680 3305




2500

3000

3500

Wav

e vel

ocity

(ms

)




150 200 250 300 350 400100Frequency (kHz)

0

10

20

30

40

50

60

70

Sens

itivi

ty (d

B)








006

004

002

0

ndash002

ndash004

ndash006

Am

plitu

de (V

)

0 1 2 3 4 5 6 7Time (s)

Hanning window

times10ndash5

(a)

A2A2


60

50

40

30

20

10

Spec

tral

ampl

itude

A1

A2

(b)



000

001

002

003

004

005

006

007

008

Dam

age i

ndic

ator

γ

I1I2I3


A 1A

2

σ = 371kPa

200 250 300 350 400 450150Voltage (V)

0000

0002

0004

0006

0008

0010





5 Conclusions






Data Availability




Acknowledgments



004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ




004

005

006

007

008

009

Dam

age i

ndic

ator

γ





References




















































10 1984 60 336 560 1680 3305




2500

3000

3500

Wav

e vel

ocity

(ms

)




150 200 250 300 350 400100Frequency (kHz)

0

10

20

30

40

50

60

70

Sens

itivi

ty (d

B)








006

004

002

0

ndash002

ndash004

ndash006

Am

plitu

de (V

)

0 1 2 3 4 5 6 7Time (s)

Hanning window

times10ndash5

(a)

A2A2


60

50

40

30

20

10

Spec

tral

ampl

itude

A1

A2

(b)



000

001

002

003

004

005

006

007

008

Dam

age i

ndic

ator

γ

I1I2I3


A 1A

2

σ = 371kPa

200 250 300 350 400 450150Voltage (V)

0000

0002

0004

0006

0008

0010





5 Conclusions






Data Availability




Acknowledgments



004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ




004

005

006

007

008

009

Dam

age i

ndic

ator

γ





References





















































006

004

002

0

ndash002

ndash004

ndash006

Am

plitu

de (V

)

0 1 2 3 4 5 6 7Time (s)

Hanning window

times10ndash5

(a)

A2A2


60

50

40

30

20

10

Spec

tral

ampl

itude

A1

A2

(b)



000

001

002

003

004

005

006

007

008

Dam

age i

ndic

ator

γ

I1I2I3


A 1A

2

σ = 371kPa

200 250 300 350 400 450150Voltage (V)

0000

0002

0004

0006

0008

0010





5 Conclusions






Data Availability




Acknowledgments



004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ




004

005

006

007

008

009

Dam

age i

ndic

ator

γ





References


















































5 Conclusions






Data Availability




Acknowledgments



004

005

006

007

008

009

010

011

Dam

age i

ndic

ator

γ




004

005

006

007

008

009

Dam

age i

ndic

ator

γ





References

















































References
















































Characterization of Contact-Type Defects in Mortar Using a ...

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Transcript of Characterization of Contact-Type Defects in Mortar Using a ...