Review of nonlinear ultrasonic guided wave nondestructive ...

Review of nonlinear ultrasonic guidedwave nondestructive evaluation:theory, numerics, and experiments

Vamshi Krishna ChillaraCliff J. Lissenden

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Review of nonlinear ultrasonic guided wavenondestructive evaluation: theory, numerics, andexperiments

Vamshi Krishna Chillara and Cliff J. Lissenden*The Pennsylvania State University, Department of Engineering Science and Mechanics, University Park, Pennsylvania 16802, United States

Abstract. Interest in using the higher harmonic generation of ultrasonic guided wave modes for nondestructiveevaluation continues to grow tremendously as the understanding of nonlinear guided wave propagation hasenabled further analysis. The combination of the attractive properties of guided waves with the attractive proper-ties of higher harmonic generation provides a very unique potential for characterization of incipient damage,particularly in plate and shell structures. Guided waves can propagate relatively long distances, provide accessto hidden structural components, have various displacement polarizations, and provide many opportunities formode conversions due to their multimode character. Moreover, higher harmonic generation is sensitive to chang-ing aspects of the microstructures such as to the dislocation density, precipitates, inclusions, and voids. Wereview the recent advances in the theory of nonlinear guided waves, as well as the numerical simulationsand experiments that demonstrate their utility. © 2015 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.55.1.011002]

Keywords: nonlinear ultrasonics; guided waves; higher harmonic generation.

Paper 150688SSV received May 25, 2015; accepted for publication Jul. 10, 2015; published online Aug. 19, 2015.

1 IntroductionNonlinear ultrasonic nondestructive evaluation uses interrog-ation signals at frequencies other than the excitationfrequency to detect changes in structural integrity and char-acterize degradation of materials. Nonlinear ultrasonic meth-odologies provide improved sensitivity to damage and theability to identify incipient damage relative to linear meth-ods. In general, linear ultrasonic methods provide good sen-sitivity to macroscale damage such as long fatigue cracks.However, in many cases, once damage appears at the macro-scale, the remaining life is short, severely limiting mainte-nance decisions. Thus, characterization of incipientdamage could facilitate a paradigm shift in operations ofstructural systems from schedule-based to condition-basedmaintenance that would ultimately enhance safety andreduce life cycle costs. Nonlinear ultrasonics is a broad dis-cipline1,2 which encompasses many specialized techniquesreliant upon nonlinear material behavior to detect and/orcharacterize incipient damage. Some of these include nonlin-ear resonant ultrasound spectroscopy,3 nonlinear elasticwave spectroscopy,4–6 and second-harmonic generation.7

Modeling relies upon both classical and nonclassical8 non-linear effects such as hysteresis. Early investigations focusedon using higher harmonic generation for characterizingmaterial microstructure.9–11 The contributions of elastic non-linearity and dislocations were examined for bulk waves.12–14

Many studies over the years have employed second-har-monic generation to characterize microstructural changes,for example, associated with fatigue, creep, or thermalaging. A recent article15 provides a thorough review of sec-ond-harmonic generation measurements. The overwhelmingmajority of the work reviewed in Refs. 1, 2, and 15 involves

bulk waves, and to a lesser extent, Rayleigh surface waves.However, over the course of the last 15 to 20 years, nonlinearultrasonic guided waves have emerged as a powerful tool forcharacterization of incipient damage in structures comprisedplates, pipes/tubes, rods, and rails. The advantages ofnonlinear guided waves are the union of the advantages ofnonlinear ultrasonics already described (i.e., improved sen-sitivity and capability to detect incipient damage) and guidedwaves (e.g., volumetric coverage, long propagation distan-ces, single-sided access, inspection speed, and inspectionof inaccessible domains). Further advantages could be real-ized by implementing noncontact methods such as laserexcitation and laser Doppler vibrometer measurements.However, due to the dispersive multimodal character ofguided waves, they present a number of analytical challengesthat bulk waves do not, and the likelihood of performing asuccessful inspection with nonlinear guided waves withoutunderstanding their propagation is just above nil.

In this article, theoretical modeling of nonlinear guidedwave propagation is summarized first in Sec. 2. Doing soenables intelligent selection of primary wave modes thatwill generate strong internally resonant higher harmonicsthrough interaction with the nonlinear elastic waveguide.Numerical simulations provide a means to test assumptionsmade in the model development and enable demonstration ofnonlinear wave propagation features without the additionalnonlinearities introduced by laboratory instrumentation.Thus, results of finite element analyses are reviewed inSec. 3. Experiments provide the only real proof of the verac-ity of the theory and simulations. Thus, Sec. 4 reviewsresults from higher harmonic generation experiments onwaveguide structures. These experimental results demonstrate

*Address all correspondence to: Cliff J. Lissenden, E-mail: [email protected] 0091-3286/2015/$25.00 © 2015 SPIE

Optical Engineering 011002-1 January 2016 • Vol. 55(1)

Optical Engineering 55(1), 011002 (January 2016) REVIEW


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the unique wave propagation characteristic of internally res-onant higher harmonics; their cumulative nature. They alsoshow how the higher harmonics generation changes as thematerial microstructure changes. The article closes with asummary and discussion of where the current state-of-knowledge could lead.

2 TheoryIn Sec. 1, we provided the motivation for using nonlinearguided waves for microscale damage detection and charac-terization. However, the mathematical complexity associatedwith guided wave propagation does not easily lend itself foranalysis from a theoretical standpoint. A first step in thisregard was taken by Deng16,17 to analyze the second-harmonic guided wave generation from shear-horizontal (SH)and Rayleigh–Lamb (RL) modes in plates. The approachwas based on the use of the partial-wave technique18 toexpand the primary wave-field, and then to examine the con-ditions under which the second harmonics from partialwaves generate a propagating guided wave field. De Limaand Hamilton19 employed a different approach to analyzesecond-harmonic generation from guided waves in plates.In this section, we describe their approach to study second-harmonic guided wave generation in plates. Unlike thestudy by De Lima and Hamilton,19 we adopt a displacementgradient-based formulation that enables a systematic analysisprocedure. Next, we introduce some notation that is used inthe rest of the article.

2.1 Notation

We used bold letters to denote vectors and tensors and denotethe position of the material particle in the reference and cur-rent configurations20 by X and x, respectively. The displace-ment of the material particles is denoted by u. Thedeformation gradient is denoted by F and is given by

EQ-TARGET;temp:intralink-;e001;63;361F ¼ ∂x∂X

: (1)

Likewise, the displacement gradient is denoted by H and isgiven by

EQ-TARGET;temp:intralink-;e002;63;298H ¼ ∂u∂X

¼ F − I; (2)

where I is the identity tensor. We use a Lagrangian measureof strain denoted by E and given by

EQ-TARGET;temp:intralink-;e003;63;235E ¼ 12ðFTF − IÞ ¼ 1

2ðHþHT þHTHÞ: (3)

Also, the linearized strain that does not include geometricnonlinearity is denoted by Elin and is given by

EQ-TARGET;temp:intralink-;e004;63;171Elin ¼12ðHþHTÞ: (4)

In this article, two equivalent (widely used) weakly non-linear hyperelastic constitutive models that describe thestrain energy function of the material are used and aregiven by

1. Landau–Lifshitz model

EQ-TARGET;temp:intralink-;e005;326;741WðEÞ ¼ 1

2λ½trðEÞ�2 þ μtrðE2Þ þ 1

3C½trðEÞ�3

þ BtrðEÞtrðE2Þ þ 1

3AtrðE3Þ: (5)

2. Murnaghan model

EQ-TARGET;temp:intralink-;e006;326;660WðEÞ¼1

2λ½trðEÞ�2þμtrðE2Þþ1

3ðlþ2mÞ½trðEÞ�3

−mtrðEÞf½trðEÞ�2− trðE2ÞgþndetðEÞ: (6)

Here, λ, μ are the Lame’s constants, A, B, C are called thethird order elastic constants, and l, m, n are called theMurnaghan constants. The constants ðA;B; CÞ and ðl; m; nÞare related by21 l ¼ Bþ C, m ¼ ð1∕2ÞAþ B, and n ¼ A.Next, we introduce some stress measures used in this article.First, the second Piola–Kirchhoff stress tensor (TRR) isobtained from WðEÞ and is given by

EQ-TARGET;temp:intralink-;e007;326;504TRR ¼ ∂WðEÞ∂E

: (7)

The first Piola–Kirchhoff stress (S) is related to TRR by

EQ-TARGET;temp:intralink-;e008;326;451S ¼ FTRR: (8)

For the Landau–Lifshitz model, the second Piola–Kirchhoffstress tensor is written in terms of strain as

EQ-TARGET;temp:intralink-;e009;326;398TRRðEÞ ¼ λtrðEÞ þ 2μEþ C½trðEÞ�2Iþ BtrðE2ÞIþ 2BtrðEÞEþ AE2: (9)

Since our theoretical formulation is based on the displace-ment gradient, we treat first and second Piola–Kirchhoffstress tensors as explicit functions of displacement gradient(H) and they are denoted as SðHÞ and TRRðHÞ. TRRðHÞ canbe obtained by using Eqs. (9) and (3) and is given by

EQ-TARGET;temp:intralink-;e010;326;297

TRRðHÞ¼ λ

2trðHþHTÞþμðHþHTÞþ λ

2trðHTHÞI

þCtrðHÞ2IþμHTHþBtrðHÞðHþHTÞ

þB2trðH2þHTHÞIþA

4ðH2þHTHþHHTþHT2Þ;

(10)

up to second order in H. Further, we decompose TRRðHÞinto two parts, namely TL

RRðHÞ and TNLRRðHÞ such that

TRRðHÞ ¼ TLRRðHÞ þ TNL

RRðHÞ. As indicated in the notation,TLRRðHÞ is linear in H and TNL

RRðHÞ is nonlinear in H and isexplicitly given by


Chillara and Lissenden: Review of nonlinear ultrasonic guided wave nondestructive evaluation. . .



TLRRðHÞ¼λ

2trðHþHTÞþμðHþHTÞ;

TNLRRðHÞ¼λ

2trðHTHÞIþCtrðHÞ2IþμHTHþBtrðHÞðHþHTÞ

þB2trðH2þHTHÞIþA

4ðH2þHTHþHHTþHT2Þ:

(11)

Likewise, using S ¼ FTRR, we can write SðHÞ ¼ SLðHÞ þSNLðHÞ whereEQ-TARGET;temp:intralink-;sec2.1;63;630SLðHÞ ¼ TL

RRðHÞ;

and

EQ-TARGET;temp:intralink-;e012;63;587SNLðHÞ ¼ HTLRRðHÞ þ TNL

RRðHÞ: (12)

Equivalent expressions can be obtained for the Murnaghanmodel. However, we restrict ourselves to the Landau–Lifshitz model in this section.

2.2 Second-Harmonic Guided Waves in Plates

Consider the schematic of the traction-free plate in the refer-ence configuration as shown in Fig. 1. We begin with thereferential form of the balance of the linear momentumgiven by

EQ-TARGET;temp:intralink-;e013;63;456Div½SðHÞ� ¼ ρκu; Snκ ¼ 0 on X2 ¼ �h; (13)

where nκ denotes the unit normal to the surface of the plate inthe reference configuration and ρκ denotes the density of thematerial in the reference configuration.

Suppose that the displacement associated with the pri-mary wave propagating in the plate is denoted by u1ðX; tÞ,and that associated with the secondary wavefield is denotedby u2ðX; tÞ, then the total displacement in the material isgiven by

EQ-TARGET;temp:intralink-;e014;63;337uðX; tÞ ¼ u1ðX; tÞ þ u2ðX; tÞ with ku2k ≪ ku1k; (14)

where the perturbation assumption is indicated. Likewise forthe displacement gradient, we have

EQ-TARGET;temp:intralink-;e015;63;284H ¼ H1 þH2 with kH2k ≪ kH1k; (15)

whereH1 ¼ ð∂u1∕∂XÞ andH2 ¼ ð∂u2∕∂XÞ are the displace-ment gradients associated with primary and secondary dis-placements. Next, we obtain the expression for the first-Piola Kirchhoff stress that goes into Eq. (13). Using Eq. (12),we getEQ-TARGET;temp:intralink-;e016;63;203

SðHÞ ¼ SLðHÞ þ SNLðHÞ ⇒SðH1 þH2Þ ¼ SLðH1 þH2Þ þ SNLðH1 þH2Þ ⇒SðH1 þH2Þ ¼ SLðH1Þ þ SLðH2Þ þ SNLðH1 þH2Þ; (16)

where the linearity of SLðHÞ was used.

As we are interested in the solution for the second har-monic, we retain only the terms of the second degree inH1 in the expression for SNLðH1 þH2Þ, and denote thoseterms by SNLðH1;H1; 2Þ which correspond to self-interac-tion22 of the primary mode. From Eq. (16), we have

EQ-TARGET;temp:intralink-;e017;326;697SðHÞ ¼ SLðH1Þ þ SLðH2Þ þ SNLðH1;H1; 2Þ: (17)

Substituting Eqs. (14) and (17) in Eq. (13), we obtain twoseparate boundary value problems for u1 and u2 as follows:

EQ-TARGET;temp:intralink-;e018;326;644Div½SLðH1Þ� − ρκu1 ¼ 0 SLðH1Þnκ ¼ 0; (18)


Div½SLðH2Þ� − ρκu2 ¼ −Div½SNLðH1;H1; 2Þ�SLðH2Þnκ ¼ −SNLðH1;H1; 2Þnκ: (19)

Now, assume u1ðX; tÞ ¼ Refu1ðX2ÞeiðkX1−ωtÞg, a propa-gating guided wave mode in the plate (can either be RLor SH mode), where Refg denotes the real part of the argu-ment, ω denotes the angular frequency, and k denotes thewavenumber of the mode. The first problem in Eq. (18) isidentically satisfied due to our assumption that u1 is a propa-gating mode in the plate. On the other hand, the solution foru2 is obtained using the normal mode expansion technique.18

Following De Lima and Hamilton,19 we seek asymptoticexpansions of SLðH2Þ and _u2 as follows:

EQ-TARGET;temp:intralink-;e020;326;456SLðH2Þ ¼Xm¼∞

m¼1

AmðX1ÞSm; _u2 ¼Xm¼∞

m¼1

AmðX1Þvm; (20)

where Smm¼∞m¼1 and vmm¼∞

m¼1 denote the stress and velocityfields corresponding to all the guided wave modes [propa-gating and nonpropagating (evanescent)] at 2ω.

As shown in Ref. 19, AmðX1Þ satisfies the following ordi-nary differential equation [Eq. (21)] for each n such thatPmn ≠ 0

EQ-TARGET;temp:intralink-;e021;326;3414Pmn

�∂Am

∂X1

− ik�nX1

�¼ ðfsurfn þ fvoln Þ: (21)

Here,EQ-TARGET;temp:intralink-;e022;326;286

Pmn ¼−14

Zh

−h

�Smv�n þ Snv�m

4:n1

�dX2;

fsurfn ¼ −12

SNLðH1;H1; 2Þv�n:n2��h

−h

fvoln ¼ 1

2

Zh

−hDiv½SNLðH1;H1; 2Þ�:v�ndX2: (22)

For every propagating modem used in the asymptotic expan-sion, there is only one propagating mode n ¼ m such thatPmn ≠ 0 and km ¼ kn. If m corresponds to a nonpropagatingmode, then km ¼ k�n. This ensures that the solution toEq. (21) is well defined and is given byEQ-TARGET;temp:intralink-;e023;326;135

AmðX1Þ ¼−iðfsurfn þ fvoln Þ

4Pmn

�eik

�nX1 − ei2kX1

k�n − 2k

�if k�n ≠ 2k

AmðX1Þ ¼ðfsurfn þ fvoln Þ

4PmnX1 if k�n ¼ 2k: (23)

Fig. 1 Cross section of the traction-free plate.




Note that if the primary mode is a propagating mode, and ifthere exists another propagating mode n ¼ m such thatk�n ¼ kn ¼ 2k, then the amplitude Am increases linearlywith the propagation distance and is termed as a cumulativesecond harmonic. While this condition is satisfied at everyfrequency for bulk waves, only specific primary guided wavemodes generate cumulative second harmonics. The two“internal resonance” conditions that a primary mode needs tosatisfy for it to generate a cumulative second harmonic are19

1. Phase-matching condition: existence of a propagatingguided wave mode at ð2ω; 2kÞ, where ðω; kÞ is the pri-mary mode.

2. Nonzero power-flux criterion: ðfsurfn þ fvoln Þ ≠ 0 forthat mode n such that k�n ¼ kn ¼ 2k.

The above analysis does not assume the nature of the pri-mary mode and can be applied to both RL and SH modes. Toidentify the guided wave modes that satisfy both the condi-tions, one needs to analyze each of them separately as out-lined below.

• Phase-matching condition19,23–26

The phase-matching condition is satisfied if andonly if there exists a propagating guided wavemode in the plate at (2ω, 2k), where (ω, k) corre-sponds to the frequency and wavenumber of the pri-mary mode. To proceed with the analysis, we firststart with the following dispersion relations


tanðqhÞtanphÞ ¼−

4k2pqðq2− k2Þ2 ðsymmetric RL modesÞ;

tanðqhÞtanðphÞ ¼−

ðq2− k2Þ24k2pq

ðantisymmetric RL modesÞ;

qh¼ nπ2

ðshear− horizontal modesÞ: (24)

Here, p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðω∕clÞ2 − k2

pand q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðω∕ctÞ2 − k2

p,

where cl and ct are the longitudinal and transversewave speeds in the material, respectively. Thephase matching condition needs to be analyzed fordifferent combinations of primary and secondarymodes. First, consider the case of a primary RLmode generating a secondary RL mode. If both theprimary ðω; kÞ and secondary modes (2ω, 2k) aresymmetric modes then we have

EQ-TARGET;temp:intralink-;sec2.2;63;241

tanðqhÞtanðphÞ ¼ −

4k2pqðq2 − k2Þ2 ;

tanð2qhÞtanð2phÞ ¼ −

4ð2kÞ2ð2pÞð2qÞ½ð2qÞ2 − ð2kÞ2�2 ;

where the first one is the dispersion relation for theprimary mode and the second one is the dispersionrelation for the secondary mode. One interestingobservation that needs to be made with regard tothe above relations is that the right hand sides ofboth of the equations are identical which enablesone to write

EQ-TARGET;temp:intralink-;sec2.2;326;752

tanðqhÞtanðphÞ ¼

tanð2qhÞtanð2phÞ :

Likewise, we can write relations for all possible com-binations of the primary and secondary RL modes asfollows:



tanð2qhÞtanð2phÞ

Primary symmetric ðantisymmetricÞ→ Secondary symmetric ðantisymmetricÞ;


tanð2phÞtanð2qhÞ

Primary symmetric ðantisymmetricÞ→ Secondary antisymmetric ðsymmetricÞ:

(25)

The first relation in Eq. (25) describes a primarymode generating a secondary mode of the samekind and the second relation describes a primarymode generating a secondary mode of the oppositekind. Similar relations can be written down for a pri-mary SH mode generating a secondary RL mode.Care must be taken in interpreting the above relations,especially when any of the terms in the equation are 0or ∞. A detailed analysis of the phase-matching con-dition can be found in Refs. 23 and 24. Next, we ana-lyze the nonzero-power-flux criterion.

• Nonzero power flux criterion19,22,25–27

To analyze nonzero power flux criterion, we adopta parity analysis in terms of the displacement gradient(H). First, we begin by observing that the displace-ment field and the corresponding displacement gra-dient in the plate for different modes are of thefollowing form given in Table 1. Here, S denotes asymmetric (even) function about the midplane ofthe plate and A denotes an antisymmetric (odd) func-tion about the midplane of the plate.

Table 1 Parity of the displacement and displacement gradient fordifferent primary modes.

Primary mode u1 (displacement) H1 (displacement gradient)

RL symmetric

0@S

A0

1A

24S A 0A S 00 0 0

35

RL antisymmetric

0@A

S0

1A

24A S 0S A 00 0 0

35

SH symmetric

0@ 0

0S

1A

24S A 0A S 00 0 0

35

SH antisymmetric

0@S

A0

1A

24A S 0S A 00 0 0

35




Correspondingly, one can show22,25 that the nonlinearterms SNLðH1;H1; 2Þ and Div½SNLðH1;H1; 2Þ� are of theparity indicated in Table 2.

From the above observations, one can conclude26 thefollowing:

1. fsurfn ¼ ð1∕2ÞSNLðH1;H1; 2Þv�n:n2jh−h ≠ 0 if and onlyif vn corresponds to a symmetric mode.

2. fvoln ¼ ð1∕2Þ∫ h−hDiv½SNLðH1;H1; 2Þ�:v�ndX2 ≠ 0 if

and only if vn corresponds to a symmetric mode.

Therefore, the power-flux fsurfn þ fvoln ≠ 0 if and only ifthe secondary mode is a symmetric RL mode. It shouldbe noted that this would also be the case even if the primarymode is an SH mode.26

From the conclusions obtained for both the phase-match-ing and nonzero power flux criteria, one can identify the fol-lowing list (Table 3) of guided wave modes that are capableof generating cumulative second harmonics. Here, m, ndenote arbitrary positive integers. In addition to the modeslisted in Table 3, other modes25 like the quasi-Rayleighmodes and (high-frequency) guided wave modes near thetransverse wave speeds are also capable of second-harmonicgeneration.

2.3 Mode Selection

The importance of selecting a primary mode that generates ahigher harmonic with strong internal resonance was men-tioned in Sec. 1, but it cannot be overemphasized. Internalresonance is the first consideration; starting with phasematching and then assessing the secondary modes that havenot just nonzero power flux, but significant power fluxes.28

Other important considerations include: modal excitabilitygiven the selected transducer, proximity of nearby modes,dispersion, diffraction, and attenuation. In fact, the numberof primary modes that generate internally resonant second

harmonics that are measured by currently available transduc-ers are so low that the generation of third harmonics wasanalyzed.29

2.4 Nonlinear Guided Waves in Other Waveguides

In this section, we briefly summarize some of the earlierwork concerning nonlinear guided waves in nonplate-likewaveguides. Second harmonic, sum, and difference fre-quency generation in waveguides of arbitrary cross sectionw first investigated by De Lima and Hamilton.30 The condi-tions for internal resonance were arrived at and also second-harmonic generation in cylindrical rods and shells wasnumerically demonstrated. Likewise, generalized higher har-monic generation in waveguides of arbitrary cross sectionwas investigated.31,32 Second-harmonic guided waves fromaxis-symmetric longitudinal modes in pipes were investi-gated in Ref. 33 using a large-radius asymptotic approxima-tion for wavestructures in pipes. Limits on the thickness todiameter ratio were discussed by Chillara and Lissenden.33 Itwas observed that these wavestructures for pipes convergeasymptotically to that of plates, hence, conclusions concern-ing second-harmonic generation for plates can be appropri-ately extended for pipes. Second-harmonic generation fromaxis-symmetric torsional and longitudinal modes was ana-lyzed in Ref. 34. Recently, higher order mode interactionsin pipes were studied35,36 where harmonic generation fromflexural modes was analyzed as well.

3 Numerical SimulationsIn this section, we discuss numerical simulations pertainingto nonlinear guided wave propagation in waveguides.Numerical simulations offer a convenient way to investigatethe cumulative harmonic generation in waveguides and offerkey insights into aspects of guided wave mode selection forefficient harmonic generation from a practical standpoint.The numerical studies enable us to alienate the nonlineareffects arising out of instrumentation and just study the effectof material nonlinearity on the wave propagation. Bothsemianalytical and numerical methods have successfullybeen employed to investigate nonlinear guided waves inwaveguides.

Nucera and Lanza di Scalea37 developed a COMSOLbased nonlinear semianalytical finite element (CO.NO.SAFE) to analyze synchronism conditions for mode-selec-tion and also to determine the modal amplitude content atsecond harmonics in waveguides. The method was demon-strated for variety of waveguides like rail, composite lami-nates, reinforced concrete slab, and so on. Finite-difference-time-domain method incorporating material and geometric

Table 2 Parity of the SNLðH1;H1;2Þ and Div½SNLðH1;H1; 2Þ�.

Primary mode SNLðH1;H1;2Þ Div½SNLðH1;H1;2Þ�

RL symmetric orRL antisymmetric

24S A 0A S 00 0 0

35

24S A 0A S 00 0 0

35

SH symmetric orSH antisymmetric

24S A 0A S 00 0 0

35

24S A 0A S 00 0 0

35

Table 3 List of guided wave modes that can generate cumulative second harmonics.

Primary mode Cut-off modes cp ¼ cl cp ¼ ffiffiffi2

pct Mode-intersections Special modes

RL symmetric (ω) nπcth

nπcl cth

ffiffiffiffiffiffiffiffiffiffic2l −c

2t

pffiffi2

pnπcl ct

hffiffiffiffiffiffiffiffiffiffiffiffi2c2t −c

2l

p All —

RL antisymmetric (ω) ð2nþ1Þπct2 h —

ffiffi2

pnπcl ct

hffiffiffiffiffiffiffiffiffiffiffiffi2c2t −c

2l

p All —

SH (ω) nπct2h

nπcl ct2h

ffiffiffiffiffiffiffiffiffiffic2l −c

2t

p — — Modes that satisfy�

1c2l− 1

c2p

�¼ m2

n2

�1c2t− 1

c2p

�




nonlinearities was employed38 to study center-frequencydependence of cumulative harmonic generation in isotropicplates. Likewise, two local approaches, namely cellularautomata finite element and local interaction simulationapproach have been employed39 to study second-harmonicguided wave generation in waveguides. On the other hand,conventional finite element methods incorporating materialand geometric nonlinearities were extensively used26,28,36,40–42

to study second-harmonic guided wave propagation in wave-guides, especially in plates and pipes. In this section, we dis-cuss the results from finite element simulations concerningsecond-harmonic guided waves in plates. First, we discuss sec-ond-harmonic guided wave generation in homogeneous, iso-tropic plates in Sec. 3.1, and then discuss results for second-harmonic guided waves in plates with inhomogeneous/local-ized nonlinearities in Sec. 3.2.

3.1 Second-Harmonic Guided Waves inHomogeneous Isotropic Plates

All the results presented in this section are obtained using thecommercial finite element software COMSOL. Results arepresented for an aluminum plate (1-mm thick) whosematerial properties are shown in Table 4. Figure 2 showsthe schematic of the model used for simulations. Thewave excitation is specified as a displacement boundary con-dition at x ¼ 0 to excite the appropriate mode. Throughoutthis section, we denote the x-component of the displacementwith “u” and the y-component of the displacement with “v.”Figure 3 shows the dispersion curves for the plate along withthe primary modes used for the study in red.

3.1.1 Cumulative versus noncumulativesecond-harmonic generation

Here, we discuss and compare the second-harmonic gener-ation from two different primary modes; S0 mode (0.5 MHz)and the S1 mode (3.6 MHz). For the FE discretization, tri-angular plane-strain elements with a maximum size of0.1 mm are employed to discretize the domain along thewave propagation direction and a minimum of 15 elementsare used along the thickness direction. A maximum time-stepof 0.01 μs is used for the S0 mode and 0.005 μs is used forthe S1 mode. Displacement amplitudes of 1 × 10−7 m and2 × 10−8 m are used for the boundary conditions for theS0 mode and the S1 mode, respectively. This choice ensuresa stress wave of a few MPa—typical of an ultrasonic wavepropagating in the material. Note while the S1 mode(3.6 MHz) satisfies the conditions of internal resonance

discussed in Sec. 2.2, while the S0 mode (0.5 MHz) doesnot satisfy the phase matching criterion of Sec. 2.2. Thephase velocity of the primary S0 mode (0.5 MHz) is5.34 mm∕μs and that of the second harmonic S0 mode(1 MHz) is 5.27 mm∕μs. On the other hand, the phase veloc-ity of both the primary S1 mode (3.6 MHz) and the secondaryS2 mode (7.2 MHz) is 6.17 mm∕μs.

Figure 4 shows the amplitude of the second harmonicfrom the S0 mode (0.5 MHz) and Fig. 5 shows the samefor the S1 mode (3.6 MHz) as a function of the normalizedpropagating distance. The normalization is carried out usingthe corresponding wavelength of the primary mode (λS0 ¼10.68 mm, λS1 ¼ 1.71 mm). Clearly, the second harmonicfrom the S0 mode (0.5 MHz) is not cumulative as it startsto decrease after about (ðx∕λS0Þ ¼ 10). On the other hand,the second harmonic from the S1 mode (3.6 MHz) is cumu-lative and increases linearly as shown in Fig. 5. This isin agreement with the prediction from the perturbationapproach presented in Sec. 2.2 that the S1 mode (3.6 MHz)generates a cumulative second harmonic.

3.1.2 Role of material and geometric nonlinearities

In this section, we compare the contribution of material andgeometric nonlinearity to the second-harmonic generation.Simulations are carried out for the primary S1 mode

Fig. 2 Schematic of the model used for simulations.

Table 4 Elastic constants in GPa used for simulations.

λ μ l m n

51 26 −250 −333 −350

Fig. 3 Dispersion curves for the aluminum plate showing the primaryand secondary modes in the simulations.

Fig. 4 Second-harmonic amplitude (A2 in m) from primary S0 mode(0.5 MHz) with normalized propagation distance.




(3.6 MHz) generating a second-harmonic S2 mode(7.2 MHz) for three variants of the constitutive model inEq. (6). These are:

1. Linear elastic material (LE)—no material or geometricnonlinearities are included, i.e., l ¼ m ¼ n ¼ 0[Eq. (6)] and linearized strain, Elin ¼ ð1∕2ÞðHþHTÞis used as a strain measure.

2. Nonlinear (NL)—both material and geometric nonli-nearities are included, i.e., l ≠ 0, m ≠ 0, n ≠ 0 (valuesfrom Table 1) and Lagrangian strain (E) is used as astrain measure.

3. Geometrically nonlinear (NG)—only geometricnonlinearity is included, i.e., l ¼ m ¼ n ¼ 0 andLagrangian strain (E) is used as a strain measure.

Figure 6 shows fast Fourier transforms for LE, NL, andNG cases at x ¼ 50 mm from the left end of the plate.Clearly, the NL case has a much higher second harmonicand is about 10 times that for the NG case. The results pre-sented here indicate that the second-harmonic generation isdominated by the material nonlinearity as opposed to thegeometric nonlinearity. Hence, it can be concluded that geo-metrically linear theories incorporating material nonlinearity

for material behavior provide a very good approximation forstudying nonlinear guided waves in plates.

3.1.3 Second-harmonic generation with groupvelocity mismatch

The results presented in the previous sections considered pri-mary modes that are (almost) phase-matched to their secondharmonic. To investigate the effect of group velocity mis-match on the second-harmonic generation, simulations arerun for the primary A0 mode (0.5 MHz). The amplitudeof the displacement boundary condition is increased to10−5 m to decipher some important aspects of second-har-monic generation as outlined later. Figure 7 shows theHilbert transform (positive envelope) of the time-domain sig-nals on a log-scale at x ¼ 40, 80, and 120 mm from the leftend of the plate. Clearly, there are two distinct peaks corre-sponding to the arrival time of the pulses; the one with alarger amplitude corresponds to the primary mode andarrives later due to a smaller group velocity (2.91 mm∕μs),and the smaller one corresponds to the secondary mode andarrives earlier in time due to a larger group velocity(5.12 mm∕μs). Also, they are clearly separated with thetime-difference between them increasing with increasingpropagation distance. A few important observations canbe made in this regard:

1. The second harmonic separates from the primarymode, the hence group velocity matching is notrequired for the higher harmonic generation. Notethat the phase matching condition is not satisfiedhere. This finding, which is explained further inRef. 40, is in direct conflict with the reasonable argu-ment presented by Muller et al.25 that the primary andsecondary wave packets must travel together in orderfor energy transfer to occur. Our explanation is that thehigher harmonic is generated by the material nonli-nearity with the nonlinear surface traction fsurfn andnonlinear body force fvoln terms [Eq. (22)] actinglike distributed sources in a similar way that paramet-ric arrays use distributed nonlinearity in fluids to gen-erate directional sound beams. Thus, we believe it isthe material’s nonlinearity itself that enables the sec-ondary wave packet to propagate and be cumulativewhen the internal resonance criteria are satisfied.

Fig. 5 Second-harmonic amplitude (A2 in m) from primary S1 mode(3.6 MHz) with normalized propagation distance.

Fig. 6 Primary mode: S1 mode (3.6 MHz)—fast Fourier transforms(log-scale) obtained from time domain signals at x ¼ 50 mm forthe three cases: linear elastic material, nonlinear, and geometricallynonlinear.

Fig. 7 Primary mode: A0 mode (0.5 MHz)—Hilbert transform of timedomain signals at x ¼ 40, 80, and 120 mm.




However, if the primary and secondary waves havedifferent group velocities, due care must be taken intheir measurement.

2. The second-harmonic mode generated is the S0 mode(1 MHz) as opposed to the A0 mode (1 MHz) as isevident from the through-thickness displacement pro-files “v” in Fig. 8 at x ¼ 120 mm during the times t ¼28 to 35 μs, which is antisymmetric about the mid-plane. This is in agreement with the predictionsfrom the perturbation approach.

3. It can be concluded that the second harmonic is con-tinuously generated from the primary mode and oncegenerated, it can propagate independent of the primarymode. This can be explained with the following ration-ale. As observed here, the second harmonic separatesfrom the primary mode and propagates as a distinctpulse independent of the primary mode. Now, theresidual primary mode can again generate a secondharmonic and this process repeats with the second-har-monic pulses being separated from the primary modewhen sufficient time has elapsed—as dictated by thegroup velocity mismatch between the primary and thesecondary modes. Also, the separated harmonic canitself generate other higher harmonics. The theoryof mode interaction presented in Ref. 22 can beused to assess the nature of such a higher harmonicgeneration. It should be noted that these are muchsmaller in magnitude when compared to the primarymode and pose an enhanced difficulty in detectingthem in an experiment.

3.2 Second-Harmonic Guided Waves in Plates withInhomogeneous/Localized Nonlinearities

3.2.1 Homogeneous versus inhomogeneousnonlinearity

Here, we compare the cumulative second-harmonic genera-tion characteristics from S1 mode (3.6 MHz) for two cases,namely homogeneous and nonhomogeneous nonlinearitiesin the plate. For the homogeneous case, the Murnaghan con-stants from Table 4 were used. For the nonhomogeneous

case, the parameters λ and μ in Table 4 were used and theMurnaghan constants were varied along the wave propaga-tion direction (x). The variation is assumed to be linear,such as for a functionally graded material i.e., lðxÞ ¼−250ð1þ x∕25Þ GPa, mðxÞ ¼ −333ð1þ x∕25Þ GPa, andnðxÞ ¼ −350ð1þ x∕25Þ GPa, where “x” is in “mm” sothat the Murnaghan constants at x ¼ 50 mm are threetimes those at x ¼ 0 mm. Figure 9 shows the amplitudeof the second harmonic with normalized propagation dis-tance. Clearly, the amplitude of the second harmonic forthe inhomogeneous case is much higher due to the highernonlinearity. Also, it should be noted that the cumulative sec-ond-harmonic generation is not linear (but super-linear) asfor the homogeneous case.

3.2.2 Effect of localized through-thickness nonlinearityon cumulative second-harmonic generation

In Sec. 3.2.1, we investigated the effect of inhomogeneousnonlinearity along the wave propagation direction. Now, weinvestigate the effect of localized through-thickness nonli-nearity on the second-harmonic generation of guidedwaves, such as may be the case in a one-sided degradationprocess. Simulations are run for two primary modes, namely,the S0 mode (0.5 MHz) and the S1 mode (3.6 MHz).Localized nonlinearity in the model is obtained by varyingthe percentage through-thickness nonlinearity as indicated inthe Fig. 10, where “LE” denotes LE and “NL” denotes non-linear elastic material. The results presented in this sectionare from Ref. 41.

S0 mode (0.5 MHz). We first present the results obtainedfor second-harmonic generation from the S0 mode at0.5 MHz. Figure 11 shows the relative nonlinearity param-eter (A2∕A2

1) at x ¼ 100 mm versus the percentage through-thickness of nonlinearity. Clearly, it increases linearly withthe amount of through-thickness nonlinearity. Also, Fig. 12shows the relative nonlinearity parameter as a function of thepropagating distance for varying levels of through-thicknessnonlinearity. As can be seen, the nonlinearity parameterincreases with the propagation distance for each of the cases.Moreover, the rate of increase of the nonlinearity parameterincreases with increasing through-thickness nonlinearity.

Fig. 8 Primary mode: A0 mode (0.5 MHz)—through thickness “v ” dis-placement profiles at x ¼ 120 mm and t ¼ 28 to 35 μs indicate the S0mode.

Fig. 9 Primary mode: S1 mode (3.6 MHz)—comparison of thesecond-harmonic amplitude (A2 in m) for homogeneous and inhomo-geneous nonlinearity distribution.




Also, it should be noted that the 20%–20% case which cor-responds to 20% nonlinearity on the top and 20% nonlinear-ity on the bottom of the plate almost coincides with the casewith 40% nonlinearity on the top. Hence, it appears that thesecond-harmonic generation from the S0 mode (0.5 MHz) isindependent of the through-thickness damage location, butonly depends on the volume-fraction of the nonlinearmaterial. This is clearly evident from the plots of the normal-ized relative nonlinearity parameter shown in Fig. 13 whereeach curve is normalized with its value at x ¼ 20 mm and allof them except one (20%) coincide. This is because, for the20% case, the energy from the primary mode is transferred tothe antisymmetric mode at the second harmonic in additionto the symmetric mode. This occurs due to the asymmetry ofthe material parameters in the top and bottom surfaces of theplate in Fig. 10.

It should be noted that the second-harmonic generationfrom the S0 mode (0.5 MHz) is independent of the locationof the damage due to its uniform wavestructure through thethickness as indicated in Fig. 14.

Next, we present the results obtained for second-harmonicgeneration from the S1 mode at 3.6 MHz.

S1 mode (3.6 MHz). Simulations similar to the one for theS0 mode (0.5 MHz) are run for the S1 mode (3.6 MHz) byvarying the through-thickness nonlinearity in the plate.Simulations were run for different cases of through-thicknessnonlinearity, namely, 20%, 40%, 60%, 80%, 90%, 100%,and 20%–20%, which again corresponds to 20% through-thickness damage on the top and 20% on the bottom. Unlikethe S0 mode (0.5 MHz), the wavestructure for the S1 mode isnot uniform through the thickness as indicated in the Fig. 15,which depicts the wavestructures of both the S1 mode(3.6 MHz) and the S2 mode (7.2 MHz). Hence, we expectthe results to be different from those obtained for the S0mode (0.5 MHz).

Figure 16 shows the relative nonlinearity parameter as afunction of the propagation distance. Several observationsare made in this regard.

1. Relative nonlinearity parameter is much higher whencompared to that from the S0 mode (0.5 MHz) due tothe higher frequency and cumulative nature of the S1mode (3.6 MHz).

2. Relative nonlinearity parameter is not monotonic withthe increasing volume-fraction of the through-thick-ness damage. Hence, it can be concluded that the sec-ond-harmonic generation from the S1 mode (3.6 MHz)

Fig. 10 Schematic of the model with through-thickness nonlinearityused for the simulation.

Fig. 11 Relative nonlinearity parameter versus the percent through-thickness nonlinearity.

Fig. 12 Relative nonlinearity parameter versus the propagation dis-tance for different levels of through-thickness nonlinearity.

Fig. 13 Normalized relative nonlinearity parameter versus the per-cent through-thickness nonlinearity.

Fig. 14 S0 mode (0.5 MHz)-wavestructure.




is not exclusively dependent on the volume fraction ofthrough-thickness nonlinearity.

3. The case of 20%–20% coincides with that for 100%,hence it can be concluded that the second-harmonicgeneration is mainly due to the contribution from thematerial near the surface rather than that from the cen-tral portion (bulk) of the plate. In fact, it appears that thebulk contribution reduces the second-harmonic gener-ation a little as is evident from Fig. 16 where the20%–20% case surpasses the 100% for x > 30 mm.

From the above study, it appears that the second-harmonicgeneration from the S1 − S2 mode pair at the longitudinalwave speed is more sensitive to the surface damage andcan be used to efficiently detect and characterize it.

4 ExperimentsNaturally, experiments are a vital part of the development ofnonlinear ultrasonic guided wave-based techniques for char-acterization of material microstructure evolution. Even moreso than for linear ultrasonic guided waves, the probability ofsuccessful experiments is low unless the nonlinear guidedwave propagation characteristics are understood and usedto select modes, frequencies, and transducers that activateprimary waves that, in turn, generate strong cumulativehigher harmonics. This section starts by describing general

considerations for nonlinear ultrasonic guided wave experi-ments intended to measure the generation of higher harmon-ics. It then reviews experimental results that have beenreported in the literature.

4.1 Measurement Considerations

The first and foremost experimental consideration is that afinite amplitude near-monochromatic waveform havingexcellent clarity is actuated. Three issues are embeddedin this consideration: (1) finite amplitude is desirable inorder that the generated higher harmonics, which have farless energy than the primary wave, are measurable; (2) thetail of the frequency distribution can overwhelm the higherharmonics unless a narrow bandwidth excitation is obtained,thus a toneburst excitation having a large number of cycles istypical; and (3) high-signal clarity, or lack of distortion,reduces the noise in the frequency spectrum making thevery low amplitude higher harmonics generated by thematerial more evident. These issues apply to bulk wavesand guided waves, but because guided waves are multimo-dal, it is even more important to get as much energy as pos-sible into the selected primary wave mode at the frequencythat will generate the higher harmonic of interest. The secondconsideration, which is linked to the first, is the intrinsic non-linearity of the measurement system. A typical measurementsystem comprises: synthesizer, amplifier, cables, transmittransducer, coupling media, test material, more couplingmedia, receive transducer, cabling, preamplifier, and oscillo-scope. Matching networks and filters are also often used toimprove system performance. It is important that the signaldistortion due to the material nonlinearity dominates the dis-tortion associated with other elements in the measurementsystem. The considerations described above are always aconcern, but specific experimental setups will have addi-tional considerations. For example, the ability to separatethe transmitter and receiver by different distances enablesthe cumulative nature of higher harmonic waves to beassessed. Hence, it provides confirmation that the distortioncausing higher harmonic generation is associated with thematerial.

4.2 Transducers

Once a primary mode and the higher harmonic that it gen-erates have been selected, transducers can be chosen to

Fig. 15 Wavestructures: (a) S1 mode (3.6 MHz) and (b) S2 mode (7.2 MHz).

Fig. 16 Cumulative second harmonic from the S1 mode (0.5 MHz) forvarying levels of through-thickness damage.




transmit and receive based on mode excitability. The wave-structure (i.e., transverse resonance pattern) for the selectedprimary mode and frequency dictates the effectiveness of atransducer to actuate that mode. The proximity of othermodes to the frequency/wavenumber of the selected modeand their excitability dictates how preferentially the selectedprimary mode is actuated with respect to the other modes.Similar considerations apply to receiving the higher har-monic mode. Waves actuated from finite-size transmitterswill diffract, which was not modeled in Sec. 2. Materialattenuation was not modeled either, thus the linear cumula-tive effect of higher harmonics given by Eq. (22) for inter-nally resonant mode pairs is not attained due to diffractionand attenuation. Note that internal resonance is not requiredfor higher harmonic generation, but it is required for thehigher harmonic to be cumulative. Preferred transmittershave a large footprint to minimize diffraction, strong cou-pling between the electrical signal and resulting mechanicaldisturbance, and minimal distortion of the waveform. Manyresearchers choose to minimize distortion at the expense ofstrong coupling by choosing single crystal lithium niobateinstead of polycrystalline lead zirconate titanate (PZT) forpiezoelectric transducers. Preferential excitation of guidedwave modes can be achieved with angle beam transducersand comb (or interdigital) transducers regardless of the typeof transduction. The means of coupling the transmitter to thematerial is important because conventional gel couplantexhibits significant nonlinearity relative to solid media. Onthe receiving side, the preferred transducer is broadband sothat it can receive both the primary and the higher harmonicfrequencies without bias, but this is often not practical.

4.3 Description of Nonlinearity

Most studies of second-harmonic generation employ someversion of the nonlinearity coefficient β to describe thematerial nonlinearity. In solids, β is often called the acousticnonlinearity parameter, much to the chagrin of the nonlinearacoustics community. The use of β for nonlinearity in solidmedia originates with lossless bulk longitudinal plane wavesmodeled in one-dimensional (1-D),43 where the boundarycondition uð0; tÞ ¼ uo cos ωt results in

EQ-TARGET;temp:intralink-;sec4.3;63;296uðx; tÞ ¼ β

8ðku0Þ2xþ uo cosðkx − tÞ

−β

8ðku0Þ2x cos½2ðkx − ωtÞ�;

so clearly the amplitudes of the primary wave and the secondharmonic are

EQ-TARGET;temp:intralink-;sec4.3;63;207A1 ¼ uo; A2 ¼β

8ðku0Þ2x;

and the nonlinearity parameter can be written as

EQ-TARGET;temp:intralink-;sec4.3;63;159β ¼ 8

k2xA2

A21

:

Thus, it is common to employ the relative nonlinearityparameter, β 0 ¼ ðA2∕A2

1Þ. However, the propagation ofguided waves is not a 1-D problem, as the transverse reso-nance in the waveguide creates unique displacement profilesfor the displacement components. Thus, strictly speaking, β

is not applicable to guided waves. It is useful to consider themodal amplitude ratios, A2∕A2

1 and A3∕A31, for second and

third harmonics, respectively, which, of course, are the rel-ative second and third order nonlinearity parameters. Itshould be emphasized that these modal amplitudes implythat A1 is constant, i.e., there is no diffraction or attenuation,and the ratio is employed because it is often difficult to rep-licate the same primary wave amplitude time after time inexperiments.

4.4 Metal Plates

Nonlinear RL waves in metal plates are the major subsectionon experiments and this is subdivided by the type of trans-ducer employed: angle beam, magnetostrictive, and disc.

4.4.1 Angle beam transducers

Angle beam transducers enable preferential mode activationthrough Snells law; i.e., the wedge angle is selected by thephase velocity of the intended mode/frequency. The firstexperiments on second-harmonic generation were reportedby Deng et al.44,45 for an aluminum plate at the A2∕S2mode intersection point (see Table 3). Their results areshown in Fig. 17 and demonstrate both second-harmonicgeneration (of the S4 mode) and its cumulative nature. Wepoint out that the group velocities of the primary and secon-dary modes do not match (3.4 and 2.3 mm∕μs).

Bermes et al.46 then showed that the S1 primary mode atthe longitudinal wave speed (cp ¼ cl, see Table 2) generatesa cumulative S2 second harmonic. A diffraction-based cor-rection factor of 1∕

ffiffiffix

pis employed when assessing the

cumulative nature of the second harmonic. These authorsused the short-time-Fourier transform to create a spectro-gram onto which they superimposed the group velocitydispersion curves as shown in Fig. 18. In this case, thegroup velocities of the S1 and S2 modes are well matched(4.3 mm∕μs). Bermes et al.47 then expand their analysis toinclude the primary S2 mode, also at cp ¼ cl, and the S4 sec-ondary mode. The use of a laser interferometer to receive thewave signals has several advantages that include: flat fre-quency response, no averaging affects from a finite sizereceiver, and because it is a noncontact measurement, it iseasy to change the propagation distance.46,47

Matlack et al.48 performed a comparative study of threeinternally resonant mode pairs (S1 − S2, S2 − S4, andA2∕S2 − S4) with the result that the S1–S2 mode pair is pre-ferred for practical reasons, but since the S2 − S4 mode pairhas a higher β 0, it would be preferred given more effectiveexperimental procedures (i.e., it is difficult to preferentiallyactivate the S2 mode at cp ¼ cl). Pruell et al.

49,50 comparedthe use of an angle beam receiver with laser interferometerreception for an aluminum plate that was plasticallydeformed. They found that β 0 increased initially due to plas-tic deformation and then remained relatively constant. The β 0values obtained from the laser interferometer were lowerthan those from the angle beam receiver because the piezo-electric transducer used for the angle beam was selectedto preferentially receive the second-harmonic frequency,and therefore, partially filtered the primary wave. Pruellet al.51 showed that β 0 increases with low-cycle fatiguefor the S1 − S2 mode pair. Cycling was performed in loadcontrol and the maximum plastic strain was less than0.02 m∕m after 50 cycles, which gave an increase of 17%




in β 0. From a theoretical perspective, it is surprising that Leeet al.52 found the A1 − A2 mode pair to give a secondharmonic that increased with the propagation length. Liuet al.53 compared second-harmonic measurements of anti-symmetric modes with symmetric modes and found themto be significantly smaller, but not zero.

4.4.2 Magnetostrictive transducers

A magnetostrictive transducer (MST) consisting of a magne-tostrictive (e.g., iron-cobalt) foil, a meandering electric coil,and a permanent magnet functions similar to a comb trans-ducer, in that the coil spacing dictates the preferred wave-length. The foil is typically coupled to the plate by adhesivebonding. By orienting the permanent magnetic field bias col-linear with the electric current in the coil, SH wave modes areactivated/received, while orienting it perpendicular results inRL wave modes. Liu et al.26 showed that the internally res-onant SH3 − S4 mode pair is cumulative and detectable withMSTs. The MST transmitter was configured to send the SH3

mode at frequency f0 ¼ 2.63 MHz, while the MST receiverwas configured to receive the S4 RL mode at 2f0. Due tothe finite size of the MSTs, energy was received at boththe primary and second-harmonic frequencies. Energy wasreceived at the primary frequency because the wavefrontwas curved, thus creating a detectable RL component fromthe primary SH wave. Finite element simulations were con-ducted to demonstrate this by investigating 20 and 100 mmwide MSTs. Lissenden et al.54 showed that the third har-monic of the fundamental SH0 mode is quite sensitive toplastic deformation. In this investigation, 2024-T3 aluminumplates were plastically deformed within a reduced widthregion having lengths of L ¼ 51, 102, 229, and 457 mm.The wave propagation distance was 430 mm, thus the plasticstrain localization increased as L decreased, even though itwas reasonably uniform (i.e., 5% to 8%) over the distance L.The modal amplitude ratio, A3∕A3

1, was 4.8 times larger for auniformly deformed plate than it was for an undeformedplate. As the localization increased, the A3∕A3

1 decreased lin-early until it was indistinguishable from the undeformedplate for a localization-to-propagation distance ratio of 0.12.Furthermore, the plastic strain level was shown to have a sig-nificant effect on the modal amplitude ratio, A3∕A3

1. In therelated experiments presented by Lissenden et al.55 usingload-controlled cycling to fatigue smooth-sided plate sam-ples, the modal amplitude ratio increased by a factor of3.6 at 80% of the fatigue life relative to the pristine material.

Fig. 17 (a) Primary and second harmonic amplitudes and (b) relative nonlinearity parameter with thepropagation distance. (Reprinted from Ref. 44 with the permission of authors and AIP Publishing.)

Fig. 18 Spectrogram showing the amplitudes and time of arrival of theprimary and secondary modes. (Reprinted from Ref. 46 with permis-sions from authors and AIP Publishing.)




No fatigue cracks were visually evident. Thus, the largechange in the modal amplitude ratio has a strong potentialto represent material degradation prior to initiation of a mac-roscale crack.

4.4.3 Disc transducers

PZT disc transducers are inexpensive and easily surface-bonded to take advantage of the shear stress activated inthe adhesive associated with radial resonance of the disc.However, mode control capabilities are very limited. Honget al.56 reported β 0 for a pristine aluminum plate that appearsto increase linearly with propagation distance. However, thePZT disc actuator actuates a circular crested wave whoseprimary amplitude decreases as 1∕

ffiffiffir

pwith propagation dis-

tance, so it is likely that the results are significantly affectedby the primary wavefront spreading. Hong et al.56 also inves-tigated the effect of a small fatigue crack on β 0 with PZTdiscs and finite element modeling that includes material non-linearity as well as contact acoustic nonlinearity (CAN) fromopening and closing of the crack. The authors plotted β 0 as afunction of distance from the crack as shown in Fig. 10 ofHong et al.56 In the experiment, a 4-mm long high-cyclefatigue crack half way through the thickness of the platewas initiated in a single edge notch sample. Material plastic-ity associated with dislocation dipoles was included in themodel, but since the authors do not discuss the plasticzone size, it is implied that the material is modeled as beinghomogeneous. If this is, indeed, the case, then the resultsindicate that the CAN dominates the material nonlinearity,otherwise β 0 would not have decreased for wave pathslocated further from the crack. Hong et al.57 use second har-monics for imaging fatigue damage at a rivet hole by usingan array of disc transducers.

We note that CAN associated with breathing cracks hasbeen studied by numerous researchers using sub- and super-harmonic generation methods.58–61

4.5 Other Waveguides (Composite Plate, Pipes,Rods, and Rail)

While the majority of the nonlinear guided wave experimentswere conducted on metal plates, other nondestructive evalu-ation applications of note are briefly mentioned. Second-harmonic generation in unidirectional composite plates sub-jected to thermal and impact damage was studied in Refs. 62and 63, respectively. Li and Cho64 also measured second-har-monic generation in a pipe. Choi et al.65 used the axisymmet-ric Tð0; 1Þ mode in an Alloy 617 pipe to generate thirdharmonics that are sensitive to the fatigue-creep damagethat was present. Nucera and Lanza di Scalea66 investigatednonlinear guided waves in solids subjected to constrainedthermal expansion with a view of being able to characterizeresidual stresses due to the thermal expansion of rails.

5 ConclusionsMaterial and geometric nonlinearities distort passing guidedwaves, which cause self-interactions to generate higher har-monics at integer multiples of the excitation frequency andmutual interactions to generate combinational harmonics.Because propagating guided waves are confined to thedispersion curves, the selection of primary modes that gen-erates internally resonant higher harmonics is a critical first

step that has been enabled by theoretical modeling. Internallyresonant mode pairs are phase matched and have nonzeropower flux, and unfortunately, are quite limited in number.Numerical simulations enable virtual experiments to be con-ducted without instrument nonlinearities and collection ofdata that is difficult to acquire from physical experiments.Simulations have shown that secondary modes once gener-ated, propagate independent of the primary mode without theneed for group velocity matching. Also, it was found that theinterplay between the wavestructure of the primary mode andlocalized material degradation significantly affects the har-monic generation. Experiments have been conducted witha variety of transmitters and receivers that demonstrate therelevant features of higher harmonic guided waves: e.g.,antisymmetric second-harmonic RL modes are not cumula-tive while symmetric second-harmonic RL modes are, andthey are sensitive to various types of microstructuralevolution. Thus, nonlinear guided waves have a strongpotential for characterization of incipient damage. In orderto achieve the potential of nonlinear guided waves, itseems expedient to correlate the higher harmonic generationwith actual features of the material microstructure. Investig-ations to relate ultrasonic nonlinearity for bulk waveswith material microstructure have been reported by numer-ous researchers.12–14,43,67,68 However, these 1-D analysisefforts may not be applicable for guided waves due totheir three-dimensional nature. Recent investigations by theauthors define an asymmetry parameter for mesoscale analy-sis that can be homogenized up to the continuum level.69,70

More research along these lines should enable a correlationbetween the higher harmonic generation and the evolution ofthe microstructure, which is incipient damage. This, in turn,will enable remaining life prediction at an early point in theservice life of structural systems.

AcknowledgmentsThis material is based upon work supported by the NationalScience Foundation under Award number 1300562. VamshiChillara also acknowledges the support from the Penn StateCollege of Engineering in the form of DistinguishedTeaching Fellowship.

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3. M. Muller et al., “Nonlinear resonant ultrasound spectroscopy (NRUS)applied to damage assessment in bone,” J. Acoust. Soc. Am. 118(6),3946–3952 (2005).

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Vamshi Krishna Chillara is a PhD student (graduation, August 2015)in the Department of Engineering Science and Mechanics at PennState. He received his MS degree in engineering mechanics fromPenn State in 2012. His research interests include ultrasonic sensingfor nondestructive evaluation and structural health monitoring (SHM)of materials, nonlinear acoustics, wave propagation in multiscalematerials, and phononics. Currently, he is a student member ofSPIE and IEEE.

Cliff J. Lissenden is a professor of engineering science and mechan-ics at Penn State. He joined the Department of Engineering Scienceand Mechanics in 1995 and added an appointment in acoustics in2011. He is an ASME fellow and was a founder of the Ben FranklinCenter of Excellence in SHM, which is the primary area of his research.Within SHM, his areas of specialization are guided wave ultrasonicsand mechanical behavior of materials.




http://dx.doi.org/10.1016/j.ijengsci.2015.04.008

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