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Mixed third harmonic shear horizontal wave generation: interactionbetween primary shear horizontal wave and second harmonic LambwaveTo cite this article: Shengbo Shan and Li Cheng 2019 Smart Mater. Struct. 28 085042

 

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Mixed third harmonic shear horizontal wavegeneration: interaction between primaryshear horizontal wave and second harmonicLamb wave

Shengbo Shan1,2,3 and Li Cheng1,2,3

1Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong2 The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen 518057, People’sRepublic of China3Hong Kong Branch of National Rail Transit Electrification and Automation Engineering TechnologyResearch Center, The Hong Kong Polytechnic University, Kowloon, Hong Kong

E-mail: li.cheng@polyu.edu.hk

Received 14 January 2019, revised 24 April 2019Accepted for publication 7 May 2019Published 23 July 2019

AbstractThe third harmonic shear horizontal (SH) waves in a weakly nonlinear plate provide newpossibilities for incipient damage detections. The understanding of their generation mechanism,however, is limited to the intuitive process of cubic self-interaction of the primary SH waves. Byconsidering both the third order and fourth order elastic constants, this paper reports the discovery ofa new generation mechanism, referred to as mixed generation, which results from the mutualinteraction between the primary SH waves and their induced second harmonic Lamb waves.Compared with linearly cumulative third harmonic SH waves induced by the cubic self-interaction ofthe primary SH waves, the mixed third harmonic SH wave amplitude also increases with the wavepropagating distance but in a wavering manner according to the phase velocity matching condition.Upon establishing a complex-domain superposition method which allows a precise extraction of thethird harmonic responses, the significance and the propagating characteristics of the mixed thirdharmonic SH waves are numerically investigated through finite element simulations. Experimentsare then conducted with a dedicated subtraction scheme to highlight the material nonlinearity ofinterest. A gel test is designed and carried out to identify the two types of third harmonic SH wavecomponents from the measured time-domain signals. Both numerical and experimental resultsconfirm the existence and the significance of the mixed third harmonic SH generation mechanism,which may impact on further applications, especially underwater damage inspections.

Keywords: mixed third harmonic SH waves, cumulative effect, complex-domain superpositionmethod, subtraction scheme

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

Introduction

Higher-order harmonic ultrasonic guided waves, usuallyresulting from the microstructural defects in a waveguide,provide a promising and feasible means for incipient damagedetections [1, 2]. Most existing efforts focus on the secondharmonic Lamb waves (2nd Lamb waves) in weakly

nonlinear plates in both theoretical and applied perspectives[3–10]. Theoretical studies prove that the 2nd Lamb wavescan be generated if only the power flux from the primarymode to the second harmonic mode is nonzero [5, 6]. Thissuggests that, whatever the primary wave mode is (symmetricor antisymmetric Lamb waves or symmetric or antisymmetricSH waves), the second harmonic wave filed should only be

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the symmetric Lamb waves [3, 6]. Furthermore, if the phasevelocities of the primary and second wave modes (approxi-mately) match, the generated second harmonic Lamb wavesare cumulative with respect to the propagating distance,which can be further explored for damage detection applica-tions [5, 6, 11]. Restricted by these two conditions, however,only a limited number of frequency-dependent mode pairs canbe used for damage detection applications [6, 11]. Based onthese predetermined mode pairs, the second harmonic Lambwaves have been applied for the detection of various types ofstructural or material detect such as fatigue [8], thermal ageddamage [9], plastic deformation [7] and so forth.

Recently, Liu et al studied the generation and propaga-tion characteristics of the third harmonic SH waves (3rd SHwaves), pointing at new possibilities for damage detectionapplications [12–14]. In contrast to the 2nd Lamb waves, the3rd SH waves are shown to be holo-internal-resonant with theprimary SH waves (1st SH waves) at all frequencies so thattheir amplitudes will always be cumulative [12]. This fre-quency-independent feature offers tremendous flexibility forthe choice of the excitation frequency for damage detectionapplications. In addition, it was experimentally demonstratedthat the 3rd SH waves are more sensitive to incipient plasticdamage than the 2nd Lamb waves are [13], which can sig-nificantly shift the damage detection limit to the earlier stage.

As compared to the 2nd Lamb waves, however, relevantresearch on the 3rd SH waves is still in its infancy and rela-tively scarce. The lack of understanding of some major issues,at both fundamental and applied levels, bottlenecks thedamage detection applications of 3rd SH waves. Up to now,the generation of the 3rd SH waves has been solely attributedto the self-interaction of the 1st SH waves [12], although thereare indications that the influence of the secondary wave fieldsgenerated by the primary waves should be also considered[15]. Generally speaking, a full understanding of the under-lying mechanism of the third harmonic SH wave generation isstill lacking. The exploration of the topic is technicallychallenging, hampered by the obvious deficiencies in theexisting tools, either numerical or experimental. For example,in the numerical perspective, the fourth order elastic constants(FOECs) are usually neglected which might be crucial for the3rd SH wave generation [14, 16]. In the experimental per-spective, the influence of various nonlinear components in asystem may easily overwhelm the small amplitude of the 3rdSH waves induced by the material nonlinearity.

Motivated by this, this paper explores the mechanism ofthe 3rd SH wave generation. Through encompassing the full setof material parameters (third- and fourth-order elastic constants,TOECs and FOECs), the mechanism behind the third harmonicwave generation is systematically investigated. In addition tothe well-known pure third harmonic generation, resulting fromthe direct cubic self-interaction of the 1st SH waves due to theFOECs for cubic material nonlinearity, a so-called mixed thirdharmonic SH wave generation mechanism is revealed andhighlighted as the quadratic interaction between the 1st SHwaves and their generated 2nd Lamb waves induced byTOECs for the quadratic material nonlinearity. Through finiteelement (FE) simulations, the significance and the propagating

characteristics of the mixed 3rd SH waves are quantified.Specifically, a complex-domain superposition method is pro-posed, which allows the separation and characterization of thethird harmonic responses. Then, experiments are conductedwith a specially designed subtraction scheme to extract thematerial nonlinearity of interest. A gel test is then designed andcarried out to confirm the existence as well as the influence ofthe mixed 3rd SH waves. The discovered mixed third harmonicgeneration mechanism, through the involvement of the secondharmonic Lamb wave field, should have a significant impact onfurther underwater applications from the wave energy leakageperspective.

Theoretical analyses

The mechanism of the third harmonic generation by the pri-mary SH waves is first investigated from a qualitative per-spective. For the sake of simplicity in the equation derivation,SH0 waves are considered. First, the material nonlinear elasticbehaviors are discussed, paving the way for subsequentanalyses. Then, the second harmonic generation by the 1st SHwaves is briefly recalled. Finally, with the knowledge of thegenerated second harmonic wave field, the 3rd SH waves arestudied, highlighting their generation mechanism and char-acteristics, with special emphasis put on the mixed 3rd SHwaves.

Material nonlinear stress–strain relationship

The material nonlinear stress–strain relationship consideringthe fourth order expansion of the strain energy can beexpressed with the extended Landau–Lifshitz model [12], as

[ ] [ ] [ ]( [ ]) [ ] [ ]

[ ] [ ] ( [ ]) [ ]( [ ])

( )

l m= + + + ++ + ++ + +

+

tr A B tr B tr

C tr E tr E tr

F tr tr F tr Gtr

H tr

T E E E E I E E

E I E I E EE E I E E E E

E I

2 2

3

2 2 4

4 ,

1

2 2

2 3 2

2 2 2

3

where λ and μ are Lamé constants; A , B , C are the LandauTOECs while E , F , G , H are the Landau FOECs of an iso-tropic material. The operation tr() denotes the trace of a matrix.In the equation, T is the second Piola-Kirchhoff stress tensorand E the Lagrangian strain tensor. Upon omitting the geo-metric nonlinearity (GN), which will be further proven to benegligible in our cases, the Lagrangian strain retreats to Cau-chy’s strain ε and the Cauchy stress σ can be used instead ofthe second Piola-Kirchhoff stress. Details on individual com-ponents of the stress–strain relation can be found in appendix A.

For a weakly nonlinear wave generation problem, theamplitude of the higher harmonic waves is usually muchsmaller than that of the primary waves. Therefore, the per-turbation theory can be applied to decompose the stress intolinear and nonlinear parts, as

( )s s s= + a2L NL

[ ] ( )s e el m= + btr 2 2L

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Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

[ ] [ ] ( [ ])[ ] [ ]

[ ] [ ] ( [ ]) [ ]( [ ]) ( )

s e e e e ee e ee e e e e ee

= + + ++ ++ + ++

A B B C

E E

F F G

H c

I II

II

tr 2 tr tr

tr 3 tr

2 tr tr 2 tr 4 tr

4 tr 2

NL 2 2 2

3 2

2 2 2

3

in which superscript L and NL denote the linear and nonlinearterms, respectively.

The second harmonic generation

Consider plane SH0 waves propagating along the x directionin a weakly nonlinear plate, as shown in figure 1. The parti-cles of the SH0 waves move along the z direction. The SH0wave field can be described in terms of the strains and onlythe component eL

13 exists [17], as

( ) ( )e e= =i

A y k ei

A k e2 2

, 3L L LSHL ik x L

SHL ik x

13 13SHL

SHL

where the superscript L denotes the linear wave components.( )e yL

13 is the normalized strain distribution across the platethickness. For SH0 waves, the strain is uniformly distributed,as ( )e =y 1.L

13 AL is the modal participation factor which isalso called modal amplitude. kSH

L denotes the wave number ofthe linear SH0 waves while x represents the wave propagatingdistance.

When SH waves propagate in a weakly nonlinear plate,higher harmonic waves are generated. First, the second har-monic generation is considered. According to the perturbationtheory, the linear wave strains of the 1st SH waves should besubstituted into equation (2c) to calculate the nonlinearstresses which correspond to the higher harmonic generation.As only the strain component eL

13 exists in the primary wavefield, those nonlinear stress terms with an order higher than( )eL

132 can be omitted. Therefore, the remaining non-zero

stress terms write

( ) ( ) ( )s s e e= = +A B a2 4NL Q L L11 11 13

213

2

( ) ( )s s e= = B b2 4NL Q L22 22 13

2

( ) ( ) ( )s s e e= = +A B c2 , 4NL Q L L33 33 13

213

2

where the superscript Q stands for the quadratic nonlinearterms. It can be seen that only three normal nonlinear stressesexist, indicating that only the second harmonic Lamb wavescan be generated, consistent with the existing theories [5, 6].

The generated nonlinear wave field can be described withthe combination of the complex reciprocity relation and thenormal mode expansion method with the following equations[18]:

⎜ ⎟⎛⎝

⎞⎠ ( ) ( · · ) · ˆ ∣

·( )

ò

s sx¶¶

- = +

+

~ ~

~

~-

-

Px

i a x

y

a

v v y

v F

4

d

5

nn n nsurf e

dd

d

dvol

n n

n

with

[ ( ) · ( ) ( ) · ( )] · ˆ ( )ò s s= - +~ ~

-P y y y y y bv v x

1

4d , 5nn

d

d

n n n n

where d is the half thickness of the plate. ( )yvn and ( )s yn arethe velocity and stress across the thickness of the nth wavemode. The overtilde represents the complex conjugate. xn isthe wave number and an(x) is the modal amplitude of aspecific wave mode; s ,surf ve and Fvol are the external exci-tations expressed as the surface traction, velocity excitation,and volume force, respectively. In this case, the excitations togenerate the nonlinear waves can be obtained from the non-linear stresses in terms of the surface traction and volumeforce as

· ˆ ( )s s= - ay 6surf NL

· ( )s= bF . 6vol NL

Substituting equations (6a) and (6b) into equation (5a),the modal amplitude of the generated second harmonicsymmetric Lamb waves can be obtained as [5, 6]

⎧⎨⎪⎪

⎩⎪⎪ ( )

( )

( )= -

- ¹

=

~~

~

~

7

a x

A

i

k ke e k k

xe k k

2for 2

for 2

,

nQ

QLambQ

SHL

i k x ik xLambQ

SHL

i k xLambQ

SHL

2

2

SHL

LambQ

SHL

where AQ is a coefficient related to the power flux from theprimary waves to the second harmonic waves. It is worthnoting that this coefficient is only related to the materialnonlinearity through TOECs. This feature is crucial to furtheranalyses.

Equation (7) shows that if the phase velocity of the 1stSH0 wave matches with that of the 2nd Lamb waves, thegenerated second harmonic amplitude will be linearlycumulative. Otherwise, it will be bounded with respect to thewave propagating distance. Unfortunately, due to the dis-persion characteristic of Lamb waves, very few 1st SH wave-2nd Lamb wave mode pairs can satisfy the condition of phasevelocity matching [6]. Therefore, except these mode pairs atsome specific frequencies, in most cases where the phasevelocities of the 1st SH waves and 2nd Lamb waves do notmatch, the strain components of the generated 2nd Lambwaves can be generally written as

( )( ) ( )e w= -~

A y e e a, 8Q Q i k x ik x11 1

2 SHL

LambQ

Figure 1. Sketch of the plate and the coordinate system.

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Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

( )( ) ( )e w= -~

A y e e b, 8Q Q i k x ik x22 2

2 SHL

LambQ

( )( ) ( )e w= -~

A y e e c, , 8Q Q i k x ik x12 3

2 SHL

LambQ

where ( )wA y, ,Q1 ( )wA y,Q

2 and ( )wA y,Q3 are the generalized

amplitudes which encompass the Lamb wave structure (thewave field distribution across the plate thickness). From thematerial perspective, these second harmonic amplitudesare only related to the TOECs. It is worth noting the strainterm eQ

12 is generated by the propagating second harmonicLamb waves although the nonlinear stress term sQ

12 doesnot exist.

The third harmonic generation

Following the same procedure, the third harmonic wave fieldgenerated by the 1st SH0 wave is investigated. First, thenonlinear stresses are calculated. For the third harmonicgeneration problem, all the terms in the order of ( )eL

133 will be

kept. Due to the existence of the second harmonic Lambwaves, their corresponding strain components e ,Q

11 eQ22 and eQ

12

in the order of ( )eL13

2 should also be considered in the cal-culations. Based on this, the following terms corresponding tothe third harmonic generation remain, as

( ) ( )( )

s s e e e e e e= = + + +B A G

a

2 8

9

NL C Q Q L Q L13 13 11 22 13 11 13 13

3

( )s s e e= = A b, 9NL C Q L23 23 12 13

where terms with a superscript C stand for the cubic nonlinearterms. The above equations show that only third harmonicshear stresses can exist, which means the third harmonic wavefield generated by 1st SH waves should also be SH wavemodes, which is also consistent with the existing under-standing [12].

The nonlinear stress terms in equations (9a) and (9b) canbe classified into two categories, to be associated with twodifferent wave generation mechanisms. The first group onlyinvolves FOECs-induced term (the third term in the right-hand side of equation (9a)) where only the primary strain isinvolved. As the FOECs are intuitively responsible for thethird harmonic generation which has been widely accepted inthe open literature, the corresponding 3rd SH waves compo-nents are therefore referred to as the pure 3rd SH waves.Meanwhile, equations (9a) and (9b) also show the presence ofthe second group of nonlinear stress terms which contains theTOECs-induced terms, appearing in the form of a product ofthe primary shear strain and second harmonic normal strainsin the equations. The corresponding 3rd SH waves compo-nents, which have not been spotted in the literature up to now,are thus named as the mixed 3rd SH waves, which will befurther scrutinized in the following analyses.

Upon substituting the strain terms corresponding to the1st SH wave field and 2nd Lamb wave field intoequations (9a) and (9b), the nonlinear stresses can be further

expressed as

( )( ) ( )( )

( )s w w= - ++~

A y e e A y ea

, ,10

C C i k x i k k x C i k x13 1

33

3SHL

LambQ

SHL

SHL

( )( ) ( )( )s w= - +~

A y e e b, , 10C C i k x i k k x23 2

3 SHL

LambQ

SHL

where ( )wA y,C1 and ( )wA y,C

2 are related to TOECs and( )wA y,C

3 to FOECs.Then, by combining the reciprocity theory and the modal

expansion method, equation (5a) can be again used to cal-culate the amplitude of the 3rd SH waves. Considering thenonlinear stresses in equation (10), the third harmonicamplitude takes the following form:

( ) ( )( )

( )= + -

+

+~~

a x A xe A e e

A xe , 11

nC C i k x C i k k x ik x

C i k x

43

5

63

SHL

LambQ

SHL

SHC

SHL

where A C4 and A C

5 are related to TOECs and A C6 to FOECs.

Therefore, the first two terms represent the modal amplitudeof the mixed 3rd SH waves. It can be seen that the first termexhibits a linearly cumulative amplitude with respect to thepropagating distance while the second one shows a boundedamplitude. The combined effect of these two terms is awaveringly cumulative behavior of the mixed 3rd SH waveamplitude. On the contrary, the last term in equation (11)represents the pure 3rd SH waves whose amplitude is linearlycumulative.

To sum up, when the 1st SH waves propagate in aweakly nonlinear plate, the 3rd SH waves are generatedthrough both TOECs and FOECs, as sketched in figure 2. Thepure 3rd SH waves related to the FOECs are generated by thecubic self-interaction of the 1st SH waves. They are linearlycumulative due to their phase velocity matching with the 1stSH waves. By comparison, the mixed 3rd SH waves inducedby the TOECs are generated through the quadratic mutualinteraction between the 1st SH waves and their generated 2ndLamb waves. Due to the mismatch of their phase velocities atmost frequencies, the mixed 3rd SH waves show a waveringlycumulative character. Compared to the pure 3rd SH waves,the newly discovered mixed 3rd SH wave generation mech-anism is less intuitive with unknown propagating character-istics up to now. Moreover, the mixed 3rd SH waves mayimpact some SH-wave-based applications. For example, inthose underwater inspection cases, the community shares thecommon belief that SH waves are immune to energy leakageinto the surrounding liquid media, exemplified by the

Figure 2. Mechanism of the third harmonic SH wave generation.

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Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

inspections of various underwater structures. The mixed 3rdharmonic SH waves, through the implication of the secondLamb waves in their generation, may challenge this belief andcall for a meticulous reexamination of the problem. Anotherpossible consequence is that the waveringly cumulative fea-ture of the mixed 3rd SH waves may pose challenges to thecumulative-guided-wave-based damage detection methodswhich examine the slope of a line fitted from the nonlinearwave amplitude-wave propagating distance curve for damagecharacterization [11]. When using the 3rd SH waves, thewaveringly cumulative feature may, in principle, affect theline fitting accuracy for damage detection. Therefore, thisnewly discovered mechanism deserves deeper analyses andvalidations, which will be addressed hereafter.

Numerical studies

Numerical studies are carried out to investigate the char-acteristics of the mixed 3rd SH waves, in comparison withthose of the pure 3rd SH waves. First, an FE model is brieflydescribed with the justification on the FOECs of the material.After that, a complex-domain superposition method is pro-posed to extract the third harmonic responses. Finally, theproperties of the mixed 3rd SH waves in terms of theirinfluence and wave propagation characteristics areinvestigated.

Model description

An FE model is established in Abaqus/Explicit as shown infigure 3. An aluminum plate, 300 mm long and 2 mm thick, isinvestigated. Periodic boundary conditions are applied in the zdirection so that plane waves can be simulated [19]. Theexcitation is a prescribed displacement in the z direction withan 8-cycle tone burst time-domain signal at 500 kHz. Theamplitude of the displacement is set to 2 μm. Uniformlydistributed displacement across the plate thickness is imposedto match the wave structure of the expectedly generated SH0wave. The mesh size is set to 0.1 mm, which ensures morethan 20 elements per smallest wavelength of interest asrecommended in the literature [20, 21]. This results in a totalof 240 000 elements and 945 315 degrees of freedom in themodel. Convergence test has been conducted before sub-sequent analyses (not shown here). The material nonlinearelastic behaviors are introduced through the VUMAT mod-ule. The Landau TOECs of isotropic aluminum can be foundin existing literature [22] while its Landau FOECs are not

available to the best of our knowledge. Nevertheless, a relatedwork provided FOECs of an aluminum crystal in terms of 11Brugger constants [23]. Therefore, a conversion between theBrugger constants and Landau constants is conducted,through which the Landau FOECs for the isotropic aluminumare estimated. Details of the conversion and parameter esti-mation can be found in appendix B. Although the obtainedFOECs may not be perfectly precise, they are believed to bein the realistic range. The material parameters used in the FEmodels are listed in table 1.

Complex-domain superposition method for signal separationand extraction

In order to characterize the third harmonic wave field, acomplex-domain superposition method is proposed to extractthe third harmonic responses in the time domain. The methodtakes the following steps:

1. Excite the plate with a tone burst signal ( )f t and get thesystem response which should contain both the linearand third harmonic components as

( ( )) ( ( ( )) ) ( )= +R g f t g f t . 12L C1

3

2. Excite the plate again with the imaginary part of theprevious excitation ( )if t which can be obtained with theHilbert transform. The corresponding response can beexpressed as

( ( )) (( ( )) ) ( ( )) ( ( ( )) )( )

= - = -R g if t ig if t ig f t ig f t .13

L C L C2

3 3

3. Multiply i to R2 and superpose R1 and iR2. The linearresponses can be eliminated and the third harmonicresponse can be obtained as

( ( ( )) ) ( )=+

g f tR iR

2. 14C 3 1 2

This procedure, named as the complex-domain super-position method, is analogous to the superposition method weproposed for the second harmonic extraction [24].

In order to validate the proposed method, a baselinemethod is used as the benchmark for which the nonlinearresponse is obtained by calculating the difference between theresponse of a system with material nonlinearity and that of itspure linear counterpart. It is worth noting that the differencesignal contains both primary and third harmonic wave com-ponents since the third harmonic energy comes from theprimary waves. A typical numerical example is examined forvalidation purposes. For the nonlinear system, only FOECsare involved. Displacement in the z direction for the SHwaves is captured at 120 mm away from the excitation posi-tion. The overall response for the nonlinear system, domi-nated by the primary SH0 component, is shown in figure 4(a).The non-dispersive nature of the SH0 wave can be clearlyobserved as the number of cycles of the received wave

Figure 3. FE model description.

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Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

package is identical to that of the excitation. Then, the non-linear response is calculated with both the baseline methodand the complex-domain superposition method, with theircomparisons shown in figure 4(b). After carrying out the fastFourier Transform (FFT) on the windowed signals, thecorresponding spectra are obtained and illustrated in figure 4(c).It can be seen that the third harmonic components obtained fromthe two methods match well. Moreover, the primary wavecomponent is completely eliminated by the complex-domainsuperposition method, demonstrating the effectiveness of the

proposed method in extracting and characterizing the third har-monic responses.

Characteristics of the mixed 3rd SH waves

With the proposed complex-domain superposition method,the influence of the GN, TOECs, and FOECs on the thirdharmonic generation is first evaluated. Through tacticallyintroducing GN/TOECs/FOECs to the FE models, theircorresponding third harmonic responses at 120 mm from the

Figure 4. FE results for the validation of the third harmonic extraction method: (a) the overall y-displacement response for the nonlinearsystem; (b) the third harmonic responses extracted with the baseline method and the proposed complex-domain superposition method; (c) thespectra of the third harmonic responses extracted with the two methods.

Table 1. Elastic constants of the aluminum plate in the FE model (Unit: GPa).

λ μ A B C E F G H

55.27 25.95 −351.2 −140.4 −102.8 400 −406 347 72

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Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

excitation position are extracted and compared in figure 5. Itcan be seen that the influence of GN is negligible comparedwith the other two cases associated with the material non-linearity. This verifies the previous assumption of neglectingGN in the theoretical analyses. In addition, the third harmonicresponses induced by TOECs and FOECs are in the sameorder of magnitude. This confirms the existence of bothmixed and pure third harmonic SH waves and further indi-cates that the newly-discovered mixed third harmoniccomponent is as important as the pure one in terms of energylevel.

The propagating characteristics of the mixed 3rd SHwaves are then investigated through comparisons with thepure ones. Note the theoretical analyses show the existence ofa bounding term in the mixed 3rd SH waves (the second termon the right-hand side of equation (11)). As an example, thisterm is calculated for the 500 kHz excitation case and itsvariation with respect to the propagating distance is plotted infigure 6. A bounding period is defined as the distance between

the two dips as shown in figure 6. In this specific case, thebounding period is 9.1 mm.

In the FE simulations, the corresponding mixed and pure3rd SH wave responses at different locations from 20 to40 mm stepped by 2 mm, are calculated. As there are 11 time-domain third harmonic signals in each case, instead of pre-senting the time-domain signals, their amplitudes are extrac-ted with the complex wavelet transform, detailed in ourprevious work [24], and plotted with respect to the wavepropagating distance. Both the pure and mixed 3rd SH wavecases are studied with results compared in figure 7. It can beseen that the pure 3rd SH waves exhibit a clear linearlycumulative characteristic (figure 7(a)) while the mixed 3rd SHwaves are waveringly cumulative (figure 7(b)) in amplitudewith respect to the propagating distance. By further sub-tracting the fitted line from the third harmonic amplitudesfrom the FE simulations in figure 7(b), the difference isobtained in figure 8, showing a bounding behavior of themixed 3rd SH wave amplitude, with a bounding period ofaround 9 mm, in agreement with the theoretical prediction infigure 6.

To sum up, FE studies confirm the existence of the mixed3rd SH waves and establish the fact that they can be asimportant as the intuitive pure ones in terms of wave ampl-itude. Besides, the waveringly cumulative characteristic of themixed 3rd SH waves is also well validated.

Experimental validations

Experiments are finally carried out to further ascertain theexistence and the significance of the mixed 3rd SH waves. Adirect scheme is first tested. After that, a specifically designedsubtraction test scheme is proposed to mitigate the influenceof undesired nonlinear sources in the measurement system.

Experimental design

Experiments are carried out on a 2 mm thick aluminum(2024-T3) plate. Magnetostrictive Transducers (MsTs) areused for the SH wave excitation and reception with thedynamic magnetic field perpendicular to the static biasmagnetic field. The spacing of the coil elements is 5.4 mmfor the actuator and 1.8 mm for the receiver, which corre-spond to the half wavelengths of the 300 and 900 kHz SH0waves in an aluminum plate respectively. A RITEC RAM5000 SNAP system is used to excite the MsT actuators with100% output level. Meanwhile, 1024 signals capturedby the MsT sensor are averaged and recorded by theoscilloscope.

The design of the experiments follows the followinglogic. In the experiments, both 3rd SH wave generationmechanism should affect the wave generation. Specifically, toidentify the mixed mechanism, a gel test is performed byplacing a layer of gel on the SH wave propagating path. Thegel allows an effective dissipation of the Lamb waves (out-of-plane particle motion) without, in principle, affecting the SHwaves (in-plane particle motion) [25]. If the mixed 3rd

Figure 5. Influence of the GN, TOECs and FOECs on the thirdharmonic signals.

Figure 6. Characteristics of the theoretically calculated boundingterm in the mixed 3rd SH waves.

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Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

harmonic SH waves (due to the mutual interaction betweenthe 1st SH waves and their generated 2nd Lamb waves) doexist in the system, the overall third harmonic response of thesystem is expected to be changed by the presence of the gel.Otherwise, it should remain the same.

Direct test scheme

First, a direct test scheme is adopted with one pair of MsTsserving as the actuator and sensor respectively, as shown infigure 9. The plate, without gel, is excited with a 10-cycletone burst signal at 300 kHz windowed by a Hann functionand the received signal is shown in figure 10(a). Then, fol-lowing the proposed complex-domain superposition method,the imaginary counterpart of the original excitation obtainedwith the Hilbert transform is applied to the actuator again. Thethird harmonic responses are further extracted, with theresults displayed in figure 10(b). Through the FFT analysis onthe windowed signals in both figures 10(a) and (b), theirrespective spectra are obtained and compared in figure 10(c).Results show that the third harmonic amplitude obtained with

these two methods matches well. Again, this demonstrates theproposed complex-domain superposition method can effec-tively extract the third harmonic responses.

Upon the deployment of the gel on the plate, the thirdharmonic response is measured again following the sameprocess and compared with the previous result withoutgel in figure 11. It can be seen that the third harmonicresponse remain almost the same after placing the gel. Itis obvious that such a testing procedure fails to demon-strate the existence of the mixed 3rd SH waves. This isbecause the nonlinearity from the transducers and instru-ments overwhelms that of the aluminum plate, which willbe addressed in the next section. Although the mixedthird harmonic generation mechanism cannot be validatedby this experimental scenario, results demonstrate twoimportant points which are important to the further ana-lyses. First, the signals received by the MsT are mainly theSH waves. Second, the gel indeed does not affect the SHwave field.

Subtraction test scheme

To mitigate the influence of the nonlinearities from the MsTsand the measurement system, a subtraction test scheme isdesigned as illustrated in figure 12. The idea is inspired by[25] which addresses a counter-propagating SH wave mixingproblem. Two MsT actuators are used to either simulta-neously or independently generate SH waves in the platethrough two independent channels of the RITEC RAM 5000SNAP system. It is important to point out that although theangles of the MsT transducers are slightly twisted, the signalsreceived will still be mainly associated with the SH waves.This is determined by the nature of the MsTs that only sheardeformations can be generated or sensed when the dynamicmagnetic field is perpendicular to the dynamic field in suchdevices [26, 27]. The subtraction test scheme is performed intwo steps. First, both actuators 1 and 2 are simultaneouslyactivated to excite SH waves at 300 kHz waves which mix inthe plate. The corresponding received signal is denoted as T

Figure 7. Propagating characteristics of (a) pure and (b) mixed 3rd SH waves.

Figure 8. Bounding characteristics of the mixed 3rd SH waves fromthe FE results.

8

Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

Figure 9. Direct scheme to validate the existence of the mixed 3rd SH waves.

Figure 10. Signals captured in the system without gel on the plate: (a) overall response associated with the original excitation; (b) thirdharmonic response extracted with the complex-domain superposition method; (c) spectra of the overall response and the extracted thirdharmonic response.

9

Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

(1+2). Second, the two actuators are separately activatedand the corresponding responses are marked as T(1) and T(2)respectively. The difference among these signals, T(1+2)-T(1)-T(2), referred to as the difference signal, is used for ana-lyses. Since the third harmonic amplitude is proportional tothe cubic of the primary wave amplitude and the primary SHwaves generated by the two actuators superpose in the firsttest, the nonlinear responses induced by the material non-linearity will remain in the difference signal. In addition, withthe two actuators working separately, the influence of thenonlinearities at the actuating parts can be in principle mini-mized or eliminated in the difference signal. Therefore, thissubtraction test scheme is expected to mitigate the undesirednonlinearities in the system.

Following the proposed procedure, experimental resultsfor a plate without gel are shown in figure 13. The threeoriginal response signals are shown in figure 13(a). Thedifference signal is then calculated and illustrated infigure 13(b), which is very weak and noisy. A Butterworthlow-pass filter is then applied to the difference signal and theresult is shown in figure 13(c), exhibiting a clear wavepackage. After taking the FFT to the windowed wavepackage in figure 13(c), the spectrum of the differencesignal can be obtained, showing the wave components at thefundamental, second and third harmonics in figure 13(d). Itis worth noting that the third harmonic amplitude, in thiscase, is two orders of magnitude smaller than that infigure 10(c). This demonstrates that the nonlinearity of themeasurement system was indeed significant in the previousdirect tests, which is now effectively mitigated by thedesigned subtraction test scheme.

Then, the gel was placed on the wave propagating path asshown in figure 12 and the same test procedure is carried outto obtain the spectrum of the difference signal. Throughcomparing the results with and without gel in figure 14(a),two important observations can be highlighted. First, since thematerial-induced second harmonic SH waves cannot theore-tically exist, the second harmonic component is attributed to

the instrumentation nonlinearity. Second, focusing on thethird harmonic response, a dramatic decrease in its amplitudecan be observed with the deployment of the gel. Afterrepeating the tests five times with corresponding error barsplotted in figure 14(b), the decreasing trend (around 30%)after the introduction of the gel is confirmed to be persistent.As the gel can only damp the Lamb waves, the experimentallyobserved decreasing trend of the third harmonic responsesclearly demonstrates the existence of the mixed 3rd SH wavesand its theoretically predicted generation mechanism, e.g.through the mutual interaction between the 1st SH waves andthe 2nd Lamb waves. In addition, the approximately 30%decrease proves its significance as a non-negligible 3rd SHwave component.

Conclusions

In this work, the mechanism of the third harmonic SH wavegeneration in a weakly nonlinear plate is investigated. Inaddition to the well-known pure third harmonic generationmechanism, a mixed mechanism is revealed. This newlydiscovered phenomenon, in terms of significance and propa-gating characteristics, is assessed and confirmed through FEsimulations and experiments. A complex-domain super-position method is proposed to extract the third harmonicwave components from the signals. For the identification ofthe mixed third harmonic SH waves, a gel test is proposedalong with the establishment of a dedicated subtraction testscheme to mitigate undesired nonlinear sources in the mea-surement system.

Theoretical analyses show that, apart from the intuitivepure third harmonic generation caused by the cubic self-interaction of the 1st SH waves with FOECs, there existsanother important path through which the so-called mixedthird harmonic SH waves can be generated as a result of thequadratic mutual interaction between the 1st SH waves andtheir generated 2nd Lamb waves associated with onlyTOECs. The mixed 3rd SH waves exhibit waveringlycumulative feature at most frequencies where the phasevelocities of the 1st SH waves and 2nd Lamb waves mis-match, in contrast to the linearly cumulative pure 3rd SHwaves. The newly discovered mixed 3rd SH wave compo-nents can be as important as their conventional pure coun-terparts in terms of amplitude and energy level, whichtherefore are a non-negligible nonlinear factor to be con-sidered in various applications.

The presented findings address the third harmonic gen-eration mechanism at the fundamental level, paving the wayfor further explorations and applications of the 3rd SH-wave-based damage detection technology. Specifically, the newlydiscovered mixed third harmonic generation mechanism mayimpact on the further applications and call for a careful sys-tem design and a meticulous handling of the diagnosis sig-nals, especially in the underwater cases considering the waveenergy leakage or those cumulative-effect-based damage

Figure 11. Comparison of the third harmonic responses before andafter the placement of gel.

10

Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

detection methods. This requires additional efforts to furtherexplore the phenomena reported in this paper and assess theirimpact on future practical applications. In that sense, thedeveloped finite element tools, the estimated FOECs and theproposed complex-domain superposition method can offer a

convenient and comprehensive package for further investi-gations of the issue. Meanwhile, the proposed subtraction testscheme to mitigate the influence of the undesired nonlinearsources can be further revamped and applied for engineeringapplications.

Figure 12. Subtraction scheme to validate the existence of the mixed third harmonic SH waves.

Figure 13. Signals captured without gel on the plate: (a) responses when the actuators are simultaneously and separately activated; (b)difference signal; (c) filtered difference signal; (d) spectrum of the difference signal.

11

Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

Acknowledgments

The project was supported by grants from the National NaturalScience Foundations of China through SHENG project (Pol-ish-Chinese Funding Initiative), the Innovation and Technol-ogy Commission of the HKSAR Government to the HongKong Branch of National Rail Transit Electrification andAutomation Engineering Technology Research Center andResearch Grants Council of Hong Kong Special Adminis-trative Region (PolyU 152070/16E). The authors would like toacknowledge Prof Zhifeng Tang from Zhejiang University forproviding us with the iron-cobalt foil made by his group.

Appendix A

The stress–strain relations for the nonlinear elastic material upto the third harmonics can be expressed as

( )

( ) ( )( )

( ) ( )( )( )

()

( )() ( )

( )( )

s l e e e me e e ee e e e e ee e e e e e ee e e e e ee e e e e e e

e e e e e e e e e e ee e e e e e

e e e e e e ee e e e e e e

e e e

= + + + + + ++ + + + + ++ + + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + + ++ + + + + +

+ + +

A.1

C

B

B A

E

E

F

F

G

H

2

2 2 2

2

3

3 3

3 3 3 3 6

2

2 2 2 2

4 2 2 2

4

11 11 22 33 11 11 22 332

112

222

332

122

132

232

11 22 33 11 112

122

132

11 22 33 112

122

132

113

223

333

11 122

11 132

22 122

33 132

22 232

33 232

12 13 23

11 22 33 112

222

332

122

132

232

11 22 332

11

112

222

332

122

132

232

11

11 22 333

( ) ( )( )( ) ( )( )( )

()

s l e e e me e e ee e e e e ee e e e e e ee e e e e ee e e e e e e

e e e e e e e e e e e

= + + + + + ++ + + + + ++ + + + + ++ + + + ++ + + + ++ + + + +

C

B

B A

E

E

2

2 2 2

2

3

3 3

3 3 3 3 6

22 11 22 33 22 11 22 332

112

222

332

122

132

232

11 22 33 22 212

222

232

11 22 33 122

222

232

113

223

333

11 122

11 132

22 122

33 132

22 232

33 232

12 13 23

( )

( )() ( )

( )( )

e e e e e ee e e e e e e

e e e e e e e

e e e

+ + + + ++ + + + + ++ + + + + +

+ + +A.2

F

F

G

H

2

2 2 2 2

4 2 2 2

4

11 22 33 112

222

332

122

132

232

11 22 332

22

112

222

332

122

132

232

22

11 22 333

( )

( ) ( )( )( ) ( )( )( )

()

( )() ( )

( )( )

s l e e e me e e ee e e e e ee e e e e e ee e e e e ee e e e e e e

e e e e e e e e e e ee e e e e e e

e e e e e ee e e e e e e

e e e

= + + + + + ++ + + + + ++ + + + + ++ + + + ++ + + + ++ + + + ++ + + + + ++ + + + ++ + + + + +

+ + +A.3

C

B

B A

E

E

F

F

G

H

2

2 2 2

2

3

3 3

3 3 3 3 6

2 2

2 2 2

4 2 2 2

4

33 11 22 33 33 11 22 332

112

222

332

122

132

232

11 22 33 33 332

232

132

11 22 33 132

232

332

113

223

333

11 122

11 132

22 122

33 132

22 232

33 232

12 13 23

11 22 33 112

222

332

122

132

232

11 22 332

33

112

222

332

122

132

232

33

11 22 333

( )

( )( ) ( )

( )( )( )

s me e e e ee e e e e e e e e ee e e e e e e e e

e e e e e e e

= + + ++ + + + + ++ + + + +

+ + + + + +

A.4

B

A F

E

G

2 2

2

3

4 2 2 2

12 12 11 22 33 12

11 12 22 12 31 32 11 22 332

12

11 22 33 11 12 22 12 31 32

112

222

332

122

132

232

12

( )

( ) ( )( )( )( )( )

s me e e e e e e e e e ee e e ee e e e e e e e e

e e e e e e e

= + + + + + ++ + ++ + + + +

+ + + + + +

A.5

B A

F

E

G

2 2

2

3

4 2 2 2

13 13 11 22 33 13 11 13 33 31 32 21

11 22 332

13

11 22 33 11 13 33 13 12 32

112

222

332

122

132

232

13

( )

( )( ) ( )

( )( )( )

s me e e e ee e e e e e e e e ee e e e e e e e e

e e e e e e e

= + + ++ + + + + ++ + + + +

+ + + + + +

A.6

B

A F

E

G

2 2

2

3

4 2 2 2 .

23 23 11 22 33 23

23 33 22 23 13 21 11 22 332

23

11 22 33 22 23 33 23 12 13

112

222

332

122

132

232

23

Figure 14. Comparison of the third harmonic responses before and after the placement of gel: (a) spectra of the difference signals; (b)amplitude of the third harmonic components with error bars.

12

Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

Appendix B

Denote the Lagrangian strain tensor as

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥( )=

E E E

E E E

E E E

E

1

2

1

21

2

1

21

2

1

2

. B.1

1 6 5

6 2 4

5 4 3

The strain energy of the an isotropic material can beexpressed with the Brugger’s model [23, 28], as

( )y y y= + +W B.22 3 4

( ) ( )

( )

( )

y = + + + + +

+ + +

c E E E c E E E E E E

c E E E

1

21

2B.3

2 11 12

22

32

12 1 2 2 3 3 1

44 42

52

62

( )

(

)

( )

(

) ( )

y = + +

+ + +

+ + + +

+ + +

+ + +

+ + + +

c E E E

c E E E E E E

E E E E E E c E E E

c E E E E E E

c E E E E E E

E E E E E E c E E E

1

61

2

1

21

2B.4

3 111 13

23

33

112 2 12

3 12

1 22

3 22

1 32

2 32

123 1 2 3

144 1 42

2 52

3 62

155 2 42

3 42

1 52

3 52

1 62

2 62

456 4 5 6

( )

( )

[ ( ) ( ) ( )]

( )

( )

( )

[ ( ) ( ) ( )]

[ ( ) ( )

( )]

( )

[ ( )]

( )

( )

y = + +

+ + + + + +

+ + +

+ + +

+ + +

+ + + + + +

+ + + +

+ +

+ + +

+ + +

+ + +

+ + +B.5

c E E E

c E E E E E E E E E

c E E E E E E

c E E E E E E E E E

c E E E E E E

c E E E E E E E E E

c E E E E E E E E

E E E E

c E E E E E E E E E

c E E E E E E

c E E E

c E E E E E E

1

241

61

41

21

41

41

2

1

2

1

241

4,

4 1111 14

24

34

1112 13

2 3 23

1 3 33

1 2

1122 12

22

22

32

32

12

1123 12

2 3 22

1 3 32

1 2

1144 12

42

22

52

32

62

1155 13

5 6 23

4 6 33

4 5

1255 1 2 42

52

2 3 62

52

1 3 42

62

1266 1 2 62

2 3 42

1 3 52

1456 4 5 6 1 2 3

4444 44

54

64

4455 42

52

52

62

42

62

where cij, cijk and the cijkl are the Brugger second, third andfourth order constants. Meanwhile, the strain energy can bealso expressed with the Murnaghan constants [2, 29], as

( )

l mm=

+- +

+- +

+ + + + B.6

W i il m

i mi i ni

p i p i i p i i p i

2

22

23

12

2 13

1 2 3

1 14

2 12

2 3 1 3 4 22

( )

( ) [( ( )) ( )] ( )= = - =

B.7

i i iE E E Etr ,1

2tr tr , det ,1 2

2 23

where l, m, n and p1, p2, p3, p4 are the Murnaghan third andfourth order constants. det() denotes the determinant. Throughcomparisons between the coefficients, the relationshipbetween the Brugger constants and Murnaghan constants canbe established as,

( )

l m

l m

m

== += + +

= + = + + += = - -

= + = - + = -

= - = - = - - -

= = = - -

= =

==

c p

c p p

c p p p

c l m c p p p p

c l c p p

c c l m n c p

c c mn

c pp

p

c c m c p p

cn

cp

c p

c p

24

24 12

24 16 16

2 4 24 20 2 16

2 2 12

2 2 2 2

22

22

4

2 2 4

4 46

2 . B.8

1111 1

1112 1 2

1122 1 2 4

111 1123 1 2 3 4

112 1144 2 3

11 123 1155 2

12 144 1255 23

4

44 155 1266 2 4

456 14563

4444 4

4455 4

13

Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

For isotropic materials, there should be only two inde-pendent SOECs, three independent TOECs and four inde-pendent FOECs. Therefore, the inherent relations among theBrugger constants for isotropic materials can be further cal-culated from equation (B8), as

The obtained relationship among the Brugger constantsagrees with the ones provided in [26], demonstrating thecorrectness of the derivation.

From the existing Brugger FOECs for aluminum crystals,11 constants are given in [23]. We take four to be theindependent ones ( =c 9916 GPa,1111 =c 3554 GPa,1155

=c 3708 GPa,1122 = -c 1000 GPa1123 ) and use them toestimate the Murnaghan FOECs for isotropic aluminummaterial. Finally, as the Landau–Lifshitz model is used in thepaper, the Landau FOECs can be calculated from the rela-tionship between the Murnaghan FOECs and Landau FOECsprovided in [29], as

( )

= + + +

= - - -

=

=

p E F G H

p E F G

p E

p G

3 2 4

3

4 . B.10

1

2

3

4

ORCID iDs

Shengbo Shan https://orcid.org/0000-0002-0950-6193Li Cheng https://orcid.org/0000-0001-6110-8099

References

[1] Jhang K Y 2009 Nonlinear ultrasonic techniques fornondestructive assessment of micro damage in material: areview Int. J. Precis. Eng. Manuf. 10 123–35

[2] Chillara V K and Lissenden C J 2015 Review of nonlinearultrasonic guided wave nondestructive evaluation: theory,numerics, and experiments Opt. Eng. 55 011002

[3] Müller M F, Kim J Y, Qu J and Jacobs L J 2010 Characteristicsof second harmonic generation of Lamb waves in nonlinearelastic plates J. Acoust. Soc. Am. 127 2141–52

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

[5] De Lima W and Hamilton M 2003 Finite-amplitude waves inisotropic elastic plates J. Sound Vib. 265 819–39

[6] Liu Y, Chillara V K and Lissenden C J 2013 On selectionof primary modes for generation of strong internallyresonant second harmonics in plate J. Sound Vib. 3324517–28

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

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

[9] Xiang Y, Deng M, Xuan F Z and Liu C J 2011 Experimentalstudy of thermal degradation in ferritic Cr–Ni alloy steelplates using nonlinear Lamb waves Ndt&E Int. 44768–74

[10] Rauter N and Lammering R 2015 Numerical simulation ofelastic wave propagation in isotropic media consideringmaterial and geometrical nonlinearities Smart Mater. Struct.24 045027

[11] Wan X, Tse P, Xu G, Tao T and Zhang Q 2016 Analytical andnumerical studies of approximate phase velocity matchingbased nonlinear S0 mode Lamb waves for the detection ofevenly distributed microstructural changes Smart Mater.Struct. 25 045023

[12] Liu Y, Chillara V K, Lissenden C J and Rose J L 2013 Thirdharmonic shear horizontal and Rayleigh Lamb waves inweakly nonlinear plates J. Appl. Phys. 114 114908

[13] Lissenden C, Liu Y, Choi G and Yao X 2014 Effect oflocalized microstructure evolution on higher harmonicgeneration of guided waves J. Nondestr. Eval. 33 178–86

[14] Liu Y, Lissenden C J and Rose J L 2014 Microstructuralcharacterization in plates using guided wave third harmonicgeneration AIP Conf. Proc. 1581 639–45

[15] Mu X, Gu X, Makarov M V, Ding Y J, Wang J, Wei J andLiu Y 2000 Third-harmonic generation by cascadingsecond-order nonlinear processes in a cerium-dopedKTiOPO4 crystal Opt. Lett. 25 117–9

[16] Liu Y 2014 Characterization of global and localized materialdegradation in plates and cylinders via nonlinear interactionof ultrasonic guided waves Dissertation The PennsylvaniaState University

( )

= -

= - +

= - + +

= + + = -

=-

= + = - -

= + = -

=

c c c

c c c c

c c c c c

c c c c c c c

cc c

c c c c c c c

c c c c c c

c c

6

88

3

10 88

36 8 4

22 2

2

3

22

31

3. B.9

1112 1111 1155

1122 1111 1155 4444

1123 1111 1155 1456 4444

111 123 144 456 1144 1155 1456

4411 12

112 123 144 1255 1155 1456 4444

155 144 456 1266 1155 4444

4455 4444

14

Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng

[17] Giurgiutiu V 2007 Structural Health Monitoring: WithPiezoelectric Wafer Active Sensors. (Amsterdam: Elsevier)

[18] Santoni G 2010 Fundamental studies in the Lamb-waveinteraction between piezoelectric wafer active sensor andhost structure during structural health monitoringDissertation University of South Carolina

[19] Wu W et al 2014 Applying periodic boundary conditions infinite element analysis SIMULIA Community Conf.,Providence

[20] Moser F, Jacobs L J and Qu J 1999 Modeling elastic wavepropagation in waveguides with the finite element methodNdt&E Int. 32 225–34

[21] Willberg C et al 2012 Comparison of different higher orderfinite element schemes for the simulation of Lamb wavesComput. Meth. Appl. Mech. Eng. 241 246–61

[22] Choi G et al 2014 Influence of localized microstructureevolution on second harmonic generation of guided wavesAIP Conf. Proc. 1581 631–8

[23] Wang H and Li M 2009 Ab initio calculations of second-, third-, and fourth-order elastic constants for single crystals Phys.Rev. B 79 224102

[24] Shan S, Cheng L and Li P 2016 Adhesive nonlinearity inLamb-wave-based structural health monitoring systemsSmart Mater. Struct. 26 025019

[25] Hasanian M and Lissenden C J 2017 Second order harmonicguided wave mutual interactions in plate: vector analysis,numerical simulation, and experimental results J. Appl.Phys. 122 084901

[26] Kim Y Y and Kwon Y E 2015 Review of magnetostrictivepatch transducers and applications in ultrasonicnondestructive testing of waveguides Ultrasonics 62 3–19

[27] Lee J S, Kim Y Y and Cho S H 2008 Beam-focused shear-horizontal wave generation in a plate by a circularmagnetostrictive patch transducer employing a planarsolenoid array Smart Mater. Struct. 18 015009

[28] Landolt H, Hellwege K, Börnstein R and Madelung O 1966Landolt-Bornstein Numerical Data and FunctionalRelationships in Science and Technology: New Series.Crystal and Solid State Physics. Group III (Berlin: Springer)

[29] Destrade M and Ogden R W 2010 On the third-and fourth-order constants of incompressible isotropic elasticityJ. Acoust. Soc. Am. 128 3334–43

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Smart Mater. Struct. 28 (2019) 085042 S Shan and L Cheng