Research Article Ultrasonic Nonlinearity Evaluation of the ...downloads.hindawi.com/journals/mpe/2016/5023127.pdf · of nondestructive testing methods to detect the imperfect condition

Research ArticleUltrasonic Nonlinearity Evaluation of the Cracked Interface

Yanjun Qiu1 Yilin Zhang1 Yewang Su2 Peng Cao3 and Youxuan Zhao4

1School of Civil Engineering Southwest Jiaotong University Chengdu 610031 China2State Key Laboratory of Nonlinear Mechanics Institute of Mechanics Chinese Academy of Sciences Beijing 100190 China3Department of Hydraulic Engineering Tsinghua University Beijing 100084 China4College of Aerospace Engineering Chongqing University Chongqing 400044 China

Correspondence should be addressed to Youxuan Zhao youxuanzhaogmailcom

Received 14 December 2015 Revised 5 May 2016 Accepted 15 May 2016

Academic Editor Roman Lewandowski

Copyright copy 2016 Yanjun Qiu et al This 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

This paper derives a novel analytical solution for acoustic nonlinearity evaluation of the cracked interface When microcracks existat the interface the tensile and compressive effective moduli of the cracked interface are considered to be different It is clearlyshown that the tension and compression elastic asymmetry can result in acoustic nonlinearity In addition numerical simulationsusing the finite element method are conducted to validate the theoretical solution It is shown that numerical results agree well withthe analytical solution Finally two factors affecting acoustic nonlinearity are studied based on the analytical solution One is thetension and compression elastic asymmetry and another is the frequency of incident wave Different from acoustic nonlinearityparameter of the general second harmonics it is found that acoustic nonlinearity parameter is a function of two factors

1 Introduction

Adhesion bonding technology for fiber reinforced compositehas been widely applied in aerospace industry Generallythe cohesive interface is very thin but can provide thepowerful strength and resiliency However the capabilityof the cohesive interface could be greatly affected by theimperfect condition such as debond fatigue damage andmicrocracks Thus there is a great need for developmentof nondestructive testing methods to detect the imperfectcondition

Nonlinear ultrasonic methods have the powerful abilityto characterize the material nonlinearity change causedby plasticity [1] imperfect interface [2ndash4] microcracks [56] and fatigue damage [7 8] When a time-harmoniclongitudinal wave propagates through a cracked solid orinterface it will cause the tension and compression elasticasymmetry then the waveform will be distorted and higher-order harmonicwaves are generated [9ndash11] And considerableexperimental evidence [12 13] has shown that ultrasonicwaves do interact with microcracks in a nonlinear fashionbut those researches may fail when the crack size is onlytens of microns which is one of the initial factors leading tointerface degradation

For the researches on acoustic nonlinearity induced bymicrocracks [14ndash16] majority are concentrated either on thescattering of elastic waves by a single crack or an arrayof cracks [17ndash20] or on the propagation of elastic wavesin a cracked medium [21ndash23] The first study on suchcontact-induced acoustic nonlinearity is probably the paperby Richardson [24] who considered the contact interfacebetween two semi-infinite half-spaces He has analyzedone-dimensional nonlinear wave propagation in a systemcomposed of an unbounded planer interface separating twosemi-infinite linear elastic media The nonlinearity is causedby the opening and closing of the interface However inRichardsonrsquos analysis the interface stiffness varying contin-uously is not accounted for Improving Richardsonrsquos theoryBiwa et al [25] have analyzed a nonlinear interface stiffnessmodel where the stiffness property of the contact interfaceis described as a function of the nominal contact pressureFurthermore by assuming the interface of the adhesive asa nonlinear spring Achenbach and Parikh [26] have inves-tigated theoretically to obtain information on the adhesivebond strength from ultrasonic test results and it is shownthat the nonlinear adhesive bond behavior could cause thegeneration of higher harmonics

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5023127 7 pageshttpdxdoiorg10115520165023127

2 Mathematical Problems in Engineering

In this paper based on the researches of Richardson andBiwa one theoretical solution for one-dimensional nonlinearwave propagation in a system composed of a multicrack-included interface separating two semi-infinite elastic mediais derived fromwhich the acoustic nonlinearity caused by thetension and compression elastic asymmetry is clearly shownTo validate the analytical solution numerical results fromthe FEM simulations are presented Comparison between theanalytical predictions and the FEM simulation results showsgood agreement Finally we also study two factors affectingacoustic nonlinearity One is the tension and compressionelastic asymmetry of the cracked interface and another is thefrequency of incident wave

2 Solution for Harmonic Wave Incidence

We consider a system composed of a cracked interfaceseparating two semi-infinite elastic media In the presenceof microcracks the interface will respond to the tensile andcompressive loadings differently which is the tension andcompression elastic asymmetry According to [11] the elasticconstants of the cracked medium can be considered to bedependent with crack density the friction of the crack facesand frequency of incident wave

1198641015840

119905

1198641015840=

1

1 + 120587119889ℎ119905

1

]1015840119905=]1015840 minus 120587119889ℎ119905

2

1 + 120587119889ℎ119905

1

1198641015840

119888

1198641015840=

1

1 + 120587119889ℎ119888

1

]1015840119888=]1015840 + 120587119888ℎ119888

1

1 + 120587119889ℎ119888

1

(1)

where1198641015840 and ]1015840 are Youngrsquosmodulus and Poissonrsquos ratio of theuncracked solid respectively And119864

120572and ]120572are respectively

the corresponding effective Youngrsquos modulus and Poissonrsquosratio of the cracked solid under tension (120572 = 119905) and undercompression (120572 = 119888) Meanwhile 1198641015840 = 119864 and ]1015840 = ] for planestress and 1198641015840 = 119864(1 minus ]2) and ]1015840 = ](1 minus ]) for plane strainThe crack density 119889 is defined as 119889 = 119873119886

2119860 where 119886 is the

average half-length of the cracks 119860 is the area of the solidcontaining 119873 randomly distributed and randomly orientedtwo-dimensional microcracks

We have

ℎ119905

1= 1 minus 119860120578

2 ln 120578 + [1205812minus 1 + 120581

4(1 minus 2 log 120581)

161205812 (1205812 minus 1)

+ 119860 (log 4 + log 120581 minus 120574)] 1205782 + 119894120587119860

21205782

ℎ119905

2= minus119861120578

2 ln 120578 + [1 + 2120574 minus 2120574120581

2minus 2 log 2

321205812

+ 119861 (log 2 minus log 120581)] 1205782 + 119894120587119861

21205782

ℎ119888

1=119865

4minus

119865 (1205814+ 1)

321205812 (1205812 minus 1)1205782 ln 120578

+

119865 [1205814+ 2 (1 + 120581

4) (log 4 minus 120574) + 2 log 120581]

641205812 (1205812 minus 1)1205782

minus1

321205871205812[tanminus1 1

120583minus

120583 (31205832+ 1)

3 (1205832 + 1)2] 1205782

+

119894120587119865 (1205814+ 1)

641205812 (1205812 minus 1)1205782

(2)

In the above 120574 is the Euler-Mascheroni constant 120578 =

119896119879119886 = 120596119886119888

119879is a dimensionless wavenumber 120581 = 119888

119871119888119879=

radic2(1 minus ])(1 minus 2]) is a dimensionless parameter 120583 is thecoefficient of friction and

119860 =5 minus 6120581

2+ 51205814

161205812 (1205812 minus 1)

119861 =1205812minus 1

161205812

119865 =2

120587(tanminus1 1

120583minus

120583

1205832 + 1)

(3)

Therefore in this paper the conclusion of [11] will beadopted and the interface can be regarded as an equivalentmedium consisting of the random distribution cracks Actu-ally this assumption has been proven in [25] for investigatingthe void inclusion medium subjected to the ultrasonic waveloading conditions

21TheGeneral Solution Firstly we assume that the elasticityof the cracked interface is a function of time and thestress in the interface is uniform In Figure 1 we illustratea one-dimensional system schematically to consider elasticlongitudinal wave propagation along 119909-axis where 119891(119909 minus 119888119905)and 119892(119909 minus 119888119905) represent the incident and transmitted wavefunctions respectively The blue area represents the crackedinterface

The two semi-infinite elastic solids are located in theregions 119909 lt 0

minus and 119909 gt 0+ respectively The equation of

responses of the interface and the stress-strain relation aredefined as follows

1205881205972119906 (119909 119905)

1205972119905=120597120590 (119909 119905)

120597119909

120590 (119909 119905) = 119864119904(119905)

120597119906 (119909 119905)

120597119909

(4)

Mathematical Problems in Engineering 3

f(x minus ct) g(x minus ct)x

h

0minus 0+

Figure 1 One-dimensional simplistic model

where 120588 is mass density 119864119904(119905) is the longitudinal stiffness of

the cracked interface 119906(119909 119905) is the displacement in the 119909-direction of an element at time 119905 from its position 119909 and120590(119909 119905) is the stress The boundary conditions in locations 0minusand 0+ are as follows

120590 (119905) = 120590 (0minus 119905) = 120590 (0

+ 119905) (5)

119906 (0+ 119905) minus 119906 (0

minus 119905) = 120590 (119905) sdot

ℎ

119864119904

(6)

where ℎ is the thickness of the cracked interfaceAccording to [24 25] the governing equation of (4)

can be solved based on travelling wave method Thereforethis paper will employ travelling wave method to obtainthe general displacement solution when longitudinal wavepropagates through the interface Then considering thecracked interface as the tension and compression elasticasymmetry this paper will achieve the corresponding specialdisplacement solution

According to the travelling wave method as a solution to(4) the following forms are considered [24 25]

119906 (119909 119905) = 119891 (119909 minus 119888119905) minus1

2119881(119905 +

119909

119888) 119909 lt 0

minus (7)

119906 (119909 119905) = 119891 (119909 minus 119888119905) +1

2119881(119905 minus

119909

119888) 119909 ge 0

+ (8)

119881 (119905) = 119906 (0+ 119905) minus 119906 (0

minus 119905) (9)

where 119888 = radic119864120588 is the wave velocity 119864 is the longitudinalstiffness of the two semi-infinite elastic solids 119891(119909 minus 119888119905)

represents the incident waves the term minus(12)119881(119905 + 119909119888)

represents reflected waves and (8) represents the transmittedwaves Equation (9) represents the equivalent displacementof the cracked interface

According to (6) and (9) we can obtain

2119881 (119905) =ℎ119864

119864119904

[2120597119891 (119909 minus 119888119905)

120597 (119909 minus 119888119905)

10038161003816100381610038161003816100381610038161003816119909=0

minus1

119888

119889119881 (119905)

119889119905] (10)

The incident wave 119891(119909 minus 119888119905) is now assumed in thefollowing form

119891 (119909 minus 119888119905) = 1198650cos [119896 (119909 minus 119888119905)] (11)

where 119896 is the wave number and 1198650is the maximum

amplitudeSubstitution of (11) in (10) yields

119889119881 (119905)

119889119905+ 120585119881 (119905) = 2119865

0120596 sin (120596119905) (12)

where 120596 is the angular frequency and 120585 = 120585(119905) = 2119888119864119904(119905)ℎ119864

Therefore the general solution of (12) is

119881 (119905) = 119890minusint 120585(119905)119889119905

int21198650120596 sin (120596119905) 119890int 120585(119905)119889119905119889119905

+ 1198621119890minusint 120585(119905)119889119905

(13)

When we know the function form of 120585(119905) we can obtainthe special solution from (13)

22 The Special Solution Assuming the different tensile andcompression effective moduli of the cracked interface from[11] the function 119864

119904(119905) can be defined by

119864119904(119905) =

119864119905

119904 when 120590 (119905) gt 0

119864119888

119904 when 120590 (119905) lt 0

(14)

where 119864119905119904and 119864

119888

119904are the tensile and compression effective

moduli respectively And the parameter 120574 = (119864119905

119904minus 119864119888

119904)119864

is introduced to represent the tension and compressionasymmetry in the elastic moduli of the cracked interface


119881 (119905)

= 21198650sin120593 [sin (120596119905

119899+ 120593) 119890

minus120585(119905minus119905119899)minus sin (120596119905 + 120593)]

(15)

where 120593 = sinminus1 (minus120596radic1205852 + 1205962)Since119881(119905

119899) = 0 and the calculation could begin from 119905

0=

0 without loss of generality the time points 119905119899+1

to switch theproperty come from the following equation

sin (120596119905119899+ 120593119905) 119890minus120585119905(119905119899+1minus119905119899)minus sin (120596119905

119899+1+ 120593119905) = 0

119899 = 0 2 4


119899+1+ 120593119888) = 0

119899 = 1 3 5

(16)


The displacement becomes

119881 (119905) =

21198650sin120593119905[sin (120596119905

119899+ 120593119905) 119890minus120585119905(119905minus119905119899)minus sin (120596119905 + 120593

119905)] 119899 = 0 2 4

21198650sin120593119888[sin (120596119905


119888)] 119899 = 1 3 5

(17)

where

120593119905= sinminus1( minus120596

radic1205852

119905+ 1205962

) 120585119905=2119888119864119905

119904

ℎ119864

120593119888= sinminus1( minus120596

radic1205852119888+ 1205962

) 120585119888=2119888119864119888

119904

ℎ119864

(18)

Substitution of (17) back to (7) and (8) gives the finialdisplacement field of the problem

3 Analytical and FEM Results

31 Analytical Results Numerical solutions using Matlab areperformed to solve (16) For a demonstrative purpose dashedline in Figure 2 shows one example of transmitted waveformsfor an incident wave with the center frequency of 1MHzobtained by numerical solutions of (16) and (17) with theparameters 120588 = 2700 kgm3 119864 = 7 times 1010 Pa 119864119905

119904= 7 times 10

9 Pa119864119888

119904= 7 times 10

10 Pa 1198650= 1 times 10

minus5m and ℎ = 1 times 10minus4m

The time-domain waveform with the dashed line is shown inFigure 2(a) and the locals in Figures 2(b1) and 2(b2) are tocompare the difference of the displacement curve subjected tothe first-half and second-half waveform in one-cycle loadingconditions It is seen that the first-half waveform in one cycleis distinctly different from the second-half waveform causedby the tension and compression elastic asymmetry AndFigure 2(c) shows the frequency spectrumusing FFTmethodit is clearly shown that the transmitted wave contains theterm with angular frequency 2120596 which is second harmonicgenerated by the cracked interface

32 Comparison with FEM Results The analytical solutionswill be validated in this section by conducting numericalsimulations of wave propagation through a cracked interfaceThe numerical simulations are performed using the finiteelementmethod (FEM) for the case of two-dimensional planestrain deformationThe commercial FEM software ABAQUSis used for this purpose

To this end a two-dimensional FEMmodel is constructedusing the four-node plane strain (CPE4R) elements Tosimulate plane waves a rectangular strip of 009m times 0003mis used in our FEM model The model consists of sim300000four-node plane strain (CPE4R) elements The user definedconstitutive law is performed to present the different tensionand compression elasticity of the cracked interface And theinterface is located in the middle of the sample with thethickness of 1times10minus4mOther regions are the isotropic elastic

material of aluminumA plane longitudinal wave is generatedfrom one end of the sampleThe amplitude longitudinal wavewith the frequency of 1MHz is 1 times 10minus5m The receiver islocated at right end of the interface

Shown in Figure 2(a) with the solid line is the FEM resultof time-domain waveform of transmitted waveThe excellentcomparison shows that the analytical solution results areidentical to that of the FEM simulation Furthermore the twoobvious amplitudes being 937 times 10minus6m and 776 times 10minus7mrespectively exist with the frequencies of ultrasonic wave at1MHz and 2MHz in Figure 2(c)

4 Acoustic Nonlinearity Parameter

The time-domain waveform curve can be obtained by theanalytical solution derived in Section 2 then the frequencyspectrum using FFT method can also be solved to obtain 119860

1

and 1198602 which are the amplitudes of the fundamental and

second harmonic respectively Through the analysis of datafrom the analytical solution we can obtain several observa-tions First 119860

2is linearly related to 119860

1 This is significantly

different from the second harmonic generated by quadraticnonlinearity in the solid where 119860

2is proportional to the

square of1198601The fundamental cause of this relationship is the

tension and compression elastic asymmetry of the crackedinterface Second 119860

2is scaled by parameter 120574 Therefore

according to [9] for acoustic nonlinearity parameter incracked solids the similar acoustic nonlinearity parameter ofthe cracked interface is introduced by

120573 =31205871198602

1198601119896ℎ (19)

For investigating the relation between the linearityparameter and the frequencies of the ultrasonic acoustica series of analytic solutions have been performed underdifferent frequencies including 05MHz 1MHz 15MHz20MHz and 25MHz Figure 3 shows the acoustic nonlin-earity parameter versus frequency It is seen that althoughthe acoustic nonlinearity parameter decreases linearly withincreasing frequency the decrease is not significant In otherwords the acoustic nonlinearity parameter has a rather weakdependence on frequency

Figure 4 shows the acoustic nonlinearity parameter ver-sus parameter 120574 It is seen that the acoustic nonlinearityparameter increases linearly with increasing 120574 which canrepresent the degree of damage for the cracked interface Inother words the degree of damage for the cracked interfacecan be characterized by the acoustic nonlinearity parameterdefined as (19)


(c) Frequency spectrum using FFT method

Am

plitu

de (m

)

0 1 2 3 4 50

02

04

06

08

1

Frequency (MHz)

times10minus5

0

2

4

6

8

10

Am

plitu

de (m

)

07 08 09 1 11 12 13 1406Time (s)

FEMAnalytical

times10minus6

times10minus6

2 3 4 5 6 7 8

0

Time (s)

minus2

minus4

minus6

minus8

Am

plitu

de (m

)

FEMAnalytical

times10minus6

times10minus7

(b1) First-half waveform in one cycle (b2) Second-half waveform in one cycle

0

05

1

15

Am

plitu

de (m

)minus05

minus11 2 3 4 50

Time (s)

FEMAnalytical

times10minus5

times10minus6

(a) Time-domain waveform

Figure 2 Waveforms (a) (b1) and (b2) and frequency spectrum (c) of transmitted wave at location 119909 = 0+


10 15 20 2505Frequency (MHz)

10

12

14

16

18

20

Acou

stic

non

linea

rity

para

met

er

Figure 3 Acoustic nonlinearity parameter versus frequency

00 02 04 0600

05

10

15

20

Acou

stic n

onlin

earit

y pa

ram

eter

120574

Figure 4 Acoustic nonlinearity parameter versus parameter 120574

5 Conclusions

As a time-harmonic longitudinal wave propagates in a systemcomposed of a cracked interface separating two semi-infiniteelastic media a second harmonic wave may be generatedby the cracked interface Assuming the longitudinal stiffnessof the interface as a function of time this paper solves theproblem and obtains the analytical general solution (anyinterface) and the special solution (the cracked interface)It is clearly shown that the tension and compression elasticasymmetry can result in acoustic nonlinearity Furthermoreto verify the developed theoretical solution we carry outdetailed numerical simulation of wave propagation througha cracked interface by using the FEM The results indicatethat the micromechanics model predictions agree well withthe FEM simulations Finally because the amplitude ofthe second harmonic is linearly related to the amplitudeof the fundamental harmonic we introduce the acousticnonlinearity parameter induced by the cracked interfacebased on the analytical solution It is shown that acousticnonlinearity parameter is a function of two factors which

are the tensioncompression elastic asymmetry and the fre-quency of incident wave respectively This study is beneficialto develop nonlinear ultrasonic quantitative NDE techniquesfor assessing microcrack-induced damage of interfaces

In future we will design and develop the reasonableexperiments to validate the solution of this paper

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Youxuan Zhao acknowledges the support from the Funda-mental Research Funds for the Central Universities (Projectno 0903005203383) Yewang Su acknowledges the supportfrom Chinese Academy of Sciences via the ldquoHundred Talentprogramrdquo and support from NSFC (no 11572323)

References

[1] A J Croxford P D Wilcox B W Drinkwater and P BNagy ldquoThe use of non-collinearmixing for nonlinear ultrasonicdetection of plasticity and fatiguerdquoThe Journal of the AcousticalSociety of America vol 126 no 5 pp EL117ndashEL122 2009

[2] J Jiao J Sun N Li G Song B Wu and C He ldquoMicro-crackdetection using a collinear wave mixing techniquerdquoNDT and EInternational vol 62 pp 122ndash129 2014

[3] S Zhou and Y Shui ldquoNonlinear reflection of bulk acousticwaves at an interfacerdquo Journal of Applied Physics vol 72 no 11pp 5070ndash5080 1992

[4] G Shui Y-S Wang P Huang and J Qu ldquoNonlinear ultrasonicevaluation of the fatigue damage of adhesive jointsrdquoNDT and EInternational vol 70 pp 9ndash15 2015

[5] V V Kazakov and A M Sutin ldquoPulsed sounding of crackswith the use of the modulation of ultrasound by vibrationsrdquoAcoustical Physics vol 47 no 3 pp 308ndash312 2001

[6] D T Zeitvogel K H Matlack J-Y Kim L J Jacobs P MSingh and J Qu ldquoCharacterization of stress corrosion crackingin carbon steel using nonlinear Rayleigh surface wavesrdquo NDTand E International vol 62 pp 144ndash152 2014

[7] P B Nagy ldquoFatigue damage assessment by nonlinear ultrasonicmaterials characterizationrdquoUltrasonics vol 36 no 1-5 pp 375ndash381 1998

[8] G P M Fierro F Ciampa D Ginzburg E Onder and MMeo ldquoNonlinear ultrasoundmodelling and validation of fatiguedamagerdquo Journal of Sound and Vibration vol 343 pp 121ndash1302015

[9] Y Zhao Y Qiu L J Jacobs and J Qu ldquoA micromechanicsmodel for the acoustic nonlinearity parameter in solids withdistributedmicrocracksrdquoAIP Conference Proceedings vol 1706Article ID 060001 2016

[10] Y Zhao Y Qiu L J Jacobs and J Qu ldquoFrequency-dependenttensile and compressive effective moduli of elastic solids withdistributed penny-shaped microcracksrdquo Acta Mechanica vol227 no 2 pp 399ndash419 2016

[11] Y Zhao Y Qiu L J Jacobs and J Qu ldquoFrequency-dependenttensile and compressive effective moduli of elastic solids with


randomly distributed two-dimensional microcracksrdquo Journal ofApplied Mechanics vol 82 no 8 Article ID 081006 2015

[12] K Jinno A Sugawara Y Ohara and K Yamanaka ldquoAnalysis onnonlinear ultrasonic images of vertical closed cracks by dampeddouble node modelrdquo Materials Transactions vol 55 no 7 pp1017ndash1023 2014

[13] J Riviere M C Remillieux Y Ohara et al ldquoDynamic acousto-elasticity in a fatigue-cracked samplerdquo Journal of NondestructiveEvaluation vol 33 no 2 pp 216ndash225 2014

[14] G Sih and J Loeber ldquoNormal compression and radial shearwaves scattering at a penny-shaped crack in an elastic solidrdquoTheJournal of the Acoustical Society of America vol 46 no 3 pp711ndash721 1969

[15] A K Mal ldquoInteraction of elastic waves with a Griffith crackrdquoInternational Journal of Engineering Science vol 8 no 9 pp763ndash776 1970

[16] A K Mal ldquoInteraction of elastic waves with a penny-shapedcrackrdquo International Journal of Engineering Science vol 8 no 5pp 381ndash388 1970

[17] J D Achenbach Ray Methods for Waves in Elastic Solids WithApplications to Scattering by Cracks Edited by A K Gautesenand H McMaken Pitman Advanced Publishing Boston MassUSA 1982

[18] J D Achenbach and A N Norris ldquoLoss of specular reflectiondue to nonlinear crack-face interactionrdquo Journal of Nondestruc-tive Evaluation vol 3 no 4 pp 229ndash239 1982

[19] Y C Angel and J D Achenbach ldquoReflection and transmissionof elastic waves by a periodic array of cracksrdquo Journal of AppliedMechanics vol 52 no 1 pp 33ndash41 1985

[20] Y C Angel and J D Achenbach ldquoReflection and transmissionof elastic waves by a periodic array of cracks oblique incidencerdquoWave Motion vol 7 no 4 pp 375ndash397 1985

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

[22] A S Eriksson A Bostrom and S K Datta ldquoUltrasonic wavepropagation through a cracked solidrdquoWave Motion vol 22 no3 pp 297ndash310 1995

[23] D Gross and C Zhang ldquoWave propagation in damaged solidsrdquoInternational Journal of Solids and Structures vol 29 no 14-15pp 1763ndash1779 1992

[24] J M Richardson ldquoHarmonic generation at an unbondedinterface-I Planar interface between semi-infinite elasticmediardquo International Journal of Engineering Science vol 17 no1 pp 73ndash85 1979

[25] S Biwa S Nakajima and N Ohno ldquoOn the acoustic nonlin-earity of solid-solid contact with pressure-dependent interfacestiffnessrdquo Journal of Applied Mechanics vol 71 no 4 pp 508ndash515 2004

[26] J D Achenbach and O K Parikh ldquoUltrasonic analysis ofnonlinear response and strength of adhesive bondsrdquo Journal ofAdhesion Science and Technology vol 5 no 8 pp 601ndash618 1991

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In this paper based on the researches of Richardson andBiwa one theoretical solution for one-dimensional nonlinearwave propagation in a system composed of a multicrack-included interface separating two semi-infinite elastic mediais derived fromwhich the acoustic nonlinearity caused by thetension and compression elastic asymmetry is clearly shownTo validate the analytical solution numerical results fromthe FEM simulations are presented Comparison between theanalytical predictions and the FEM simulation results showsgood agreement Finally we also study two factors affectingacoustic nonlinearity One is the tension and compressionelastic asymmetry of the cracked interface and another is thefrequency of incident wave

2 Solution for Harmonic Wave Incidence

We consider a system composed of a cracked interfaceseparating two semi-infinite elastic media In the presenceof microcracks the interface will respond to the tensile andcompressive loadings differently which is the tension andcompression elastic asymmetry According to [11] the elasticconstants of the cracked medium can be considered to bedependent with crack density the friction of the crack facesand frequency of incident wave

1198641015840

119905

1198641015840=

1

1 + 120587119889ℎ119905

1

]1015840119905=]1015840 minus 120587119889ℎ119905

2

1 + 120587119889ℎ119905

1

1198641015840

119888

1198641015840=

1

1 + 120587119889ℎ119888

1

]1015840119888=]1015840 + 120587119888ℎ119888

1

1 + 120587119889ℎ119888

1

(1)

where1198641015840 and ]1015840 are Youngrsquosmodulus and Poissonrsquos ratio of theuncracked solid respectively And119864

120572and ]120572are respectively

the corresponding effective Youngrsquos modulus and Poissonrsquosratio of the cracked solid under tension (120572 = 119905) and undercompression (120572 = 119888) Meanwhile 1198641015840 = 119864 and ]1015840 = ] for planestress and 1198641015840 = 119864(1 minus ]2) and ]1015840 = ](1 minus ]) for plane strainThe crack density 119889 is defined as 119889 = 119873119886

2119860 where 119886 is the

average half-length of the cracks 119860 is the area of the solidcontaining 119873 randomly distributed and randomly orientedtwo-dimensional microcracks

We have

ℎ119905

1= 1 minus 119860120578

2 ln 120578 + [1205812minus 1 + 120581

4(1 minus 2 log 120581)

161205812 (1205812 minus 1)

+ 119860 (log 4 + log 120581 minus 120574)] 1205782 + 119894120587119860

21205782

ℎ119905

2= minus119861120578

2 ln 120578 + [1 + 2120574 minus 2120574120581

2minus 2 log 2

321205812

+ 119861 (log 2 minus log 120581)] 1205782 + 119894120587119861

21205782

ℎ119888

1=119865

4minus

119865 (1205814+ 1)

321205812 (1205812 minus 1)1205782 ln 120578

+

119865 [1205814+ 2 (1 + 120581

4) (log 4 minus 120574) + 2 log 120581]

641205812 (1205812 minus 1)1205782

minus1

321205871205812[tanminus1 1

120583minus

120583 (31205832+ 1)

3 (1205832 + 1)2] 1205782

+

119894120587119865 (1205814+ 1)

641205812 (1205812 minus 1)1205782

(2)

In the above 120574 is the Euler-Mascheroni constant 120578 =

119896119879119886 = 120596119886119888

119879is a dimensionless wavenumber 120581 = 119888

119871119888119879=

radic2(1 minus ])(1 minus 2]) is a dimensionless parameter 120583 is thecoefficient of friction and

119860 =5 minus 6120581

2+ 51205814

161205812 (1205812 minus 1)

119861 =1205812minus 1

161205812

119865 =2

120587(tanminus1 1

120583minus

120583

1205832 + 1)

(3)

Therefore in this paper the conclusion of [11] will beadopted and the interface can be regarded as an equivalentmedium consisting of the random distribution cracks Actu-ally this assumption has been proven in [25] for investigatingthe void inclusion medium subjected to the ultrasonic waveloading conditions

21TheGeneral Solution Firstly we assume that the elasticityof the cracked interface is a function of time and thestress in the interface is uniform In Figure 1 we illustratea one-dimensional system schematically to consider elasticlongitudinal wave propagation along 119909-axis where 119891(119909 minus 119888119905)and 119892(119909 minus 119888119905) represent the incident and transmitted wavefunctions respectively The blue area represents the crackedinterface

The two semi-infinite elastic solids are located in theregions 119909 lt 0

minus and 119909 gt 0+ respectively The equation of

responses of the interface and the stress-strain relation aredefined as follows

1205881205972119906 (119909 119905)

1205972119905=120597120590 (119909 119905)

120597119909

120590 (119909 119905) = 119864119904(119905)

120597119906 (119909 119905)

120597119909

(4)



h

0minus 0+




120590 (119905) = 120590 (0minus 119905) = 120590 (0

+ 119905) (5)

119906 (0+ 119905) minus 119906 (0

minus 119905) = 120590 (119905) sdot

ℎ

119864119904

(6)




119906 (119909 119905) = 119891 (119909 minus 119888119905) minus1

2119881(119905 +

119909

119888) 119909 lt 0

minus (7)

119906 (119909 119905) = 119891 (119909 minus 119888119905) +1

2119881(119905 minus

119909

119888) 119909 ge 0

+ (8)

119881 (119905) = 119906 (0+ 119905) minus 119906 (0

minus 119905) (9)





2119881 (119905) =ℎ119864

119864119904

[2120597119891 (119909 minus 119888119905)

120597 (119909 minus 119888119905)

10038161003816100381610038161003816100381610038161003816119909=0

minus1

119888

119889119881 (119905)

119889119905] (10)


119891 (119909 minus 119888119905) = 1198650cos [119896 (119909 minus 119888119905)] (11)



119889119881 (119905)

119889119905+ 120585119881 (119905) = 2119865

0120596 sin (120596119905) (12)



119881 (119905) = 119890minusint 120585(119905)119889119905

int21198650120596 sin (120596119905) 119890int 120585(119905)119889119905119889119905

+ 1198621119890minusint 120585(119905)119889119905

(13)




119864119904(119905) =

119864119905

119904 when 120590 (119905) gt 0

119864119888

119904 when 120590 (119905) lt 0

(14)

where 119864119905119904and 119864

119888



119904minus 119864119888

119904)119864



119881 (119905)

= 21198650sin120593 [sin (120596119905

119899+ 120593) 119890


(15)



0=




119899+1+ 120593119905) = 0

119899 = 0 2 4


119899+1+ 120593119888) = 0

119899 = 1 3 5

(16)



119881 (119905) =

21198650sin120593119905[sin (120596119905


119905)] 119899 = 0 2 4

21198650sin120593119888[sin (120596119905


119888)] 119899 = 1 3 5

(17)

where

120593119905= sinminus1( minus120596

radic1205852

119905+ 1205962

) 120585119905=2119888119864119905

119904

ℎ119864

120593119888= sinminus1( minus120596

radic1205852119888+ 1205962

) 120585119888=2119888119864119888

119904

ℎ119864

(18)




119904= 7 times 10

9 Pa119864119888

119904= 7 times 10

10 Pa 1198650= 1 times 10









1











120573 =31205871198602

1198601119896ℎ (19)





Am

plitu

de (m

)

0 1 2 3 4 50

02

04

06

08

1

Frequency (MHz)

times10minus5

0

2

4

6

8

10

Am

plitu

de (m

)

07 08 09 1 11 12 13 1406Time (s)

FEMAnalytical

times10minus6

times10minus6

2 3 4 5 6 7 8

0

Time (s)

minus2

minus4

minus6

minus8

Am

plitu

de (m

)

FEMAnalytical

times10minus6

times10minus7


0

05

1

15

Am

plitu

de (m

)minus05

minus11 2 3 4 50

Time (s)

FEMAnalytical

times10minus5

times10minus6





10

12

14

16

18

20

Acou

stic

non

linea

rity

para

met

er


00 02 04 0600

05

10

15

20

Acou

stic n

onlin

earit

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ram

eter

120574


5 Conclusions




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











h

0minus 0+




120590 (119905) = 120590 (0minus 119905) = 120590 (0

+ 119905) (5)

119906 (0+ 119905) minus 119906 (0

minus 119905) = 120590 (119905) sdot

ℎ

119864119904

(6)




119906 (119909 119905) = 119891 (119909 minus 119888119905) minus1

2119881(119905 +

119909

119888) 119909 lt 0

minus (7)

119906 (119909 119905) = 119891 (119909 minus 119888119905) +1

2119881(119905 minus

119909

119888) 119909 ge 0

+ (8)

119881 (119905) = 119906 (0+ 119905) minus 119906 (0

minus 119905) (9)





2119881 (119905) =ℎ119864

119864119904

[2120597119891 (119909 minus 119888119905)

120597 (119909 minus 119888119905)

10038161003816100381610038161003816100381610038161003816119909=0

minus1

119888

119889119881 (119905)

119889119905] (10)


119891 (119909 minus 119888119905) = 1198650cos [119896 (119909 minus 119888119905)] (11)



119889119881 (119905)

119889119905+ 120585119881 (119905) = 2119865

0120596 sin (120596119905) (12)



119881 (119905) = 119890minusint 120585(119905)119889119905

int21198650120596 sin (120596119905) 119890int 120585(119905)119889119905119889119905

+ 1198621119890minusint 120585(119905)119889119905

(13)




119864119904(119905) =

119864119905

119904 when 120590 (119905) gt 0

119864119888

119904 when 120590 (119905) lt 0

(14)

where 119864119905119904and 119864

119888



119904minus 119864119888

119904)119864



119881 (119905)

= 21198650sin120593 [sin (120596119905

119899+ 120593) 119890


(15)



0=




119899+1+ 120593119905) = 0

119899 = 0 2 4


119899+1+ 120593119888) = 0

119899 = 1 3 5

(16)



119881 (119905) =

21198650sin120593119905[sin (120596119905


119905)] 119899 = 0 2 4

21198650sin120593119888[sin (120596119905


119888)] 119899 = 1 3 5

(17)

where

120593119905= sinminus1( minus120596

radic1205852

119905+ 1205962

) 120585119905=2119888119864119905

119904

ℎ119864

120593119888= sinminus1( minus120596

radic1205852119888+ 1205962

) 120585119888=2119888119864119888

119904

ℎ119864

(18)




119904= 7 times 10

9 Pa119864119888

119904= 7 times 10

10 Pa 1198650= 1 times 10









1











120573 =31205871198602

1198601119896ℎ (19)





Am

plitu

de (m

)

0 1 2 3 4 50

02

04

06

08

1

Frequency (MHz)

times10minus5

0

2

4

6

8

10

Am

plitu

de (m

)

07 08 09 1 11 12 13 1406Time (s)

FEMAnalytical

times10minus6

times10minus6

2 3 4 5 6 7 8

0

Time (s)

minus2

minus4

minus6

minus8

Am

plitu

de (m

)

FEMAnalytical

times10minus6

times10minus7


0

05

1

15

Am

plitu

de (m

)minus05

minus11 2 3 4 50

Time (s)

FEMAnalytical

times10minus5

times10minus6





10

12

14

16

18

20

Acou

stic

non

linea

rity

para

met

er


00 02 04 0600

05

10

15

20

Acou

stic n

onlin

earit

y pa

ram

eter

120574


5 Conclusions




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881 (119905) =

21198650sin120593119905[sin (120596119905


119905)] 119899 = 0 2 4

21198650sin120593119888[sin (120596119905


119888)] 119899 = 1 3 5

(17)

where

120593119905= sinminus1( minus120596

radic1205852

119905+ 1205962

) 120585119905=2119888119864119905

119904

ℎ119864

120593119888= sinminus1( minus120596

radic1205852119888+ 1205962

) 120585119888=2119888119864119888

119904

ℎ119864

(18)




119904= 7 times 10

9 Pa119864119888

119904= 7 times 10

10 Pa 1198650= 1 times 10









1











120573 =31205871198602

1198601119896ℎ (19)





Am

plitu

de (m

)

0 1 2 3 4 50

02

04

06

08

1

Frequency (MHz)

times10minus5

0

2

4

6

8

10

Am

plitu

de (m

)

07 08 09 1 11 12 13 1406Time (s)

FEMAnalytical

times10minus6

times10minus6

2 3 4 5 6 7 8

0

Time (s)

minus2

minus4

minus6

minus8

Am

plitu

de (m

)

FEMAnalytical

times10minus6

times10minus7


0

05

1

15

Am

plitu

de (m

)minus05

minus11 2 3 4 50

Time (s)

FEMAnalytical

times10minus5

times10minus6





10

12

14

16

18

20

Acou

stic

non

linea

rity

para

met

er


00 02 04 0600

05

10

15

20

Acou

stic n

onlin

earit

y pa

ram

eter

120574


5 Conclusions




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Am

plitu

de (m

)

0 1 2 3 4 50

02

04

06

08

1

Frequency (MHz)

times10minus5

0

2

4

6

8

10

Am

plitu

de (m

)

07 08 09 1 11 12 13 1406Time (s)

FEMAnalytical

times10minus6

times10minus6

2 3 4 5 6 7 8

0

Time (s)

minus2

minus4

minus6

minus8

Am

plitu

de (m

)

FEMAnalytical

times10minus6

times10minus7


0

05

1

15

Am

plitu

de (m

)minus05

minus11 2 3 4 50

Time (s)

FEMAnalytical

times10minus5

times10minus6





10

12

14

16

18

20

Acou

stic

non

linea

rity

para

met

er


00 02 04 0600

05

10

15

20

Acou

stic n

onlin

earit

y pa

ram

eter

120574


5 Conclusions




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10

12

14

16

18

20

Acou

stic

non

linea

rity

para

met

er


00 02 04 0600

05

10

15

20

Acou

stic n

onlin

earit

y pa

ram

eter

120574


5 Conclusions




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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