Title Frequency Dependence of Second-Harmonic Generation inLamb Waves

Author(s) Matsuda, Naoki; Biwa, Shiro

Citation Journal of Nondestructive Evaluation (2014), 33(2): 169-177

Issue Date 2014-02-07

URL http://hdl.handle.net/2433/194150

Right

The final publication is available at Springer viahttp://dx.doi.org/10.1007/s10921-014-0227-y.; この論文は出版社版でありません。引用の際には出版社版をご確認ご利用ください。This is not the published version. Please citeonly the published version.

Type Journal Article

Textversion author

Kyoto University

1

This article appeared as:

N. Matsuda and S. Biwa, Frequency dependence of second-harmonic generation in Lamb waves, Journal of Nondestructive Evaluation, Vol. 33 (2014), pp. 169-177.

Frequency dependence of second-harmonic generation in Lamb waves

Naoki Matsuda and Shiro Biwa*

Department of Aeronautics and Astronautics, Graduate School of Engineering,

Kyoto University, Katsura, Nishikyo-ku, Kyoto 615-8540, Japan

Abstract

The frequency dependence of the second-harmonic generation in Lamb waves is studied theoretically and numerically in order to examine the role of phase matching for sensitive evaluation of material nonlinearity. Nonlinear Lamb wave propagation in an isotropic plate is analyzed using the perturbation technique and the modal decomposition in the neighborhood of a typical frequency satisfying the phase matching. The results show that the ratio of the amplitude of second-harmonic Lamb mode to the squared amplitude of fundamental Lamb mode grows cumulatively in a certain range of fundamental frequency for a finite propagation distance. It is also shown that the frequency for which this ratio reaches maximum is close but not equal to the phase-matching frequency when the propagation distance is finite. This feature is confirmed numerically using the finite-difference time-domain method incorporating material and geometrical nonlinearities. The fact that the amplitude of second-harmonic mode becomes high in a finite range of fundamental frequency proves robustness of the material evaluation method using second harmonics in Lamb waves. Keywords: Nonlinear ultrasonics; Higher harmonic generation; Lamb wave; Perturbation analysis; Finite-Difference Time-Domain method

* Corresponding author: Shiro Biwa, E-mail biwa@kuaero.kyoto-u.ac.jp

2

1. INTRODUCTION

It is an important issue to ensure safety of structures used for industrial plants, transportation,

etc., by evaluating material degradation due to creep, fatigue and plastic deformation. Ultrasonic

waves are widely utilized for nondestructive evaluation (NDE) of such material damage.

Foregoing studies have revealed that acoustic nonlinearity of materials, commonly manifested as

higher harmonic generation, has enhanced sensitivity to material damage [1-3]. A number of

studies have been carried out to explore higher harmonic generation in Lamb wave propagation to

assess early stage of damage in plate-like structures [4-20]. In particular, Deng [4] experimentally

observed cumulative growth of harmonic amplitude with propagation distance, i.e. cumulative

harmonic generation, which is only possible under certain conditions. A perturbation

approximation and modal analysis was used by de Lima and Hamilton [5] and Deng [6] to obtain

the wave field of second harmonics in Lamb waves. They clarified that the conditions for

cumulative growth of harmonic amplitude are (1) the phase-matching condition, and (2) non-zero

power flux from the primary to the higher-harmonic Lamb wave. Deng and Pei [7] used the

cumulative second-harmonic Lamb wave to evaluate fatigue damage in plates. Pruell et al. [8]

examined the correlation between the acoustic nonlinearity measured with Lamb waves and the

level of plastic strain in aluminum and aluminum alloy plates. Xiang et al. [9] measured thermal

degradation in ferric Cr-Ni alloy steel plates using nonlinear Lamb waves. Bermes et al. [10] used

a laser interferometric detection system to develop a procedure to measure material nonlinearity.

Harmonic generation in Lamb waves of higher orders was analyzed theoretically by Srivastava

and Lanza di Scalea [11]. Müller et al. [13] found five Lamb mode types which satisfy

requirements for the cumulative second-harmonic generation, including three types in an

asymptotic sense. Matsuda and Biwa [14] showed that the Lamb modes which satisfy the exact

phase-matching condition are classified into four mode types including two types discussed by

Müller et al. [13].

The phase-matching condition plays an important role in selecting the frequency of the

fundamental mode to carry out sensitive evaluation of material nonlinearity. As demonstrated in

foregoing theoretical works [6, 20] the generation and accumulation behavior of the

second-harmonic Lamb mode can be qualitatively different depending on whether the fundamental

frequency meets the phase matching or not. Experimentally [7, 16-18], the observed

second-harmonic components have a peak at the fundamental frequency satisfying the phase

matching. In practical situations, it is though difficult to tune the fundamental frequency to meet

the phase matching exactly, due to the lack of precise acoustic properties of the plate or

experimental setting errors. At a finite propagation distance, however, substantial growth of the

second-harmonic mode, if not ideally cumulative, may be expected when the fundamental

frequency meets the phase matching approximately. Therefore, it is of practical interest to explore

the fundamental frequency dependence of harmonic generation in Lamb waves in a neighborhood

of the phase-matching frequency.

In this paper, we present a theoretical and numerical analysis of the frequency dependence of

the second-harmonic generation in Lamb waves. Nonlinear Lamb wave propagation in an isotropic

plate is analyzed using the perturbation technique and the modal decomposition. The frequency

dependence of surface displacement amplitude of the generated second-harmonic Lamb waves is

examined in the neighborhood of a typical frequency satisfying the phase matching. The frequency

depen

incorp

2. FO

2.1 GC

with

1X stress

where

refere

vecto

quant

t . In

const

2.2 L

T

decom

where

ndence is als

porating mat

ORMULATI

Governing equConsider a p

the quadrati

2X plane a

s-strain relati

e jX deno

ence configur

or, ijH the

tities ijP , U

Eq. (2), λ

tants [23] and

Linearization b

The nonlinea

mposition. Th

e L

iU is the

so analyzed

terial and geo

ON

uations for anlate of thickn

ic nonlinearit

as shown in

on and the bo

ρ

HλP kkij

ote the comp

ration, ijP t

displacemen

iU and ijH a

and μ are L

d ijδ is Kron

by perturbati

ar equations

he solution U

e primary sol

Fig.1 G

numerically

ometrical non

n isotropic plness h2 wh

ty. The plate

Fig. 1. In th

oundary cond

2

2

0

i

Xt

Uρ

,

(

(41

δHCH

HHB

HHA

HHμ

Hμδ

ijllkk

ijkk

ik

jkik

ijijk

3 iP

ponents of t

the first Piola

nt gradient te

are considere

Lamé’s elasti

necker’s delt

ion analysis

(1)-(3) are so

iU is assum

iU

lution and U

Geometry of an

3

by the finit

nlinearities.

late hich is made

e has two st

he absence o

ditions [21-2

,0

j

ij

X

P

,

)

)(

21 HH

HHH

HHH

HλH

ji

kjki

kjik

ji

,0 3X

the referenc

a-Kirchhoff s

ensor with re

ed as functio

ic constants,

ta.

olved by usi

med to be a suNLii UU

NLiU is the se

n elastic plate

te-difference

of an isotrop

tress-free sur

of body force

4] are given

,3 hXh

,

(

)

3

21

hXh

HHH

H

HH

HHH

klkl

jk

kjki

ijkk

,h

e position v

stress tensor,

espect to the

ns of the ref

A , B and

ng the pertur

um of two ter

,NL

econdary sol

with free boun

time-domain

pic and lossle

rfaces S, whi

es, the equati

by

)δ

δHH

ijlk

ijklkl

vector, 0ρ th

iU the par

e reference c

ference positi

d C are the

rbation analy

ms:

ution. When

ndaries

n (FDTD) m

ess elastic m

ich are para

ion of motio

the density i

rticle displac

configuration

ion jX and

third-order e

ysis and the

weak nonlin

method

edium

llel to

on, the

in the

ement

n. The

d time

elastic

modal

nearity

4

is assumed, the nonlinear equations (1)-(3) are divided into a linear homogenous and a linear

non-homogeneous differential equations, which are given by

,0)( LL

2

L2

0

Hijj

i PXt

Uρ ,3 hXh

,0)( LL3 HiP ,3 hX

and

),()( LNLNLL

2

NL2

0 HH iijj

i FPXt

Uρ

,3 hXh

),()( LNL3

NLL3 HH ii PP ,3 hX

where LH and NLH are the displacement gradients as to the primary solution L

iU and the

secondary solution NL

iU , respectively. In Eqs. (5)-(8), NL

iF is defined as

jiji XPF )()( LNLLNL HH and LijP and NL

ijP are the linear and the quadratic terms of

the first Piola-Kirchhoff stress tensor:

,)(Ljiijijkkij HHμδHλP H

.

)()(.

))((

)(

21

41

21NL

ijllkk

ijlkklkljiijkk

jkkjkiik

kjkikjikjkik

ijklklijkkij

δHCH

δHHHHHHB

HHHHA

HHHHHHμ

δHHHHλP

H

3. ANALYSIS OF HARMONIC GENERATION IN LAMB WAVES

3.1 Modal Expansion of Lamb waves

While Eqs. (5) and (6) have Lamb mode solutions and the horizontally polarized shear mode

solutions, we consider only the Lamb mode solutions as primary solutions. Here we consider the

l th Lamb mode with frequency 0ω as a solution of Eqs. (5) and (6). The primary solution is

given by

c.c.,)(exp)(2

1013

),(0

L 0 tωXκiXA lωκlUU

where c.c. stands for complex conjugate, and )( 3),( 0 XωκlU denotes the displacement profile of

the l th Lamb mode whose wavenumber and angular frequency are lκ and 0ω , respectively.

The displacement profile, )( 3),( 0 XωκlU , is normalized so that the displacement field

c.c.exp)(2

1013

),( 0 tωXκiX lωκlU has the unit energy flux density in the 1X direction.

The secondary solution NLU can be divided into second-harmonic term

ω2U and DC term ω0U . Now we consider only the second-harmonic term

ω2U , which is written as a linear

combination of Lamb modes:

5

,c.c.2exp)()(2

103

)2,(1

2 0 m

ωκm

ω tωiXXA mUU

where )2,( 0ωκmU denotes the displacement profile of the m th Lamb mode whose wave number

and frequency are mκ and 02ω , respectively. Suppose that the m th Lamb mode is a

propagating mode. The modal amplitude )( 1XAm is given by [5, 25],

,2,e4

,2,)2(

1ee

4)(

12

volsurf

)2(2

volsurf

1

1

1

1

lmXκi

mm

mm

lmlm

XκκiXκi

mm

mm

m

κκXP

ff

κκκκiP

ff

XA

l

lm

l

where

,

22224

131

)2,(*)2,(*)2,()2,( 0000

dXPh

h

ωκωκωκωκ

mm

mmmm

e

VσVσ

,

22

3

3

0

1

*)2,(2surf

hX

hX

ωκω

m

m

f

eVp

.

22 3

*)2,(2vol

0

h

h

ωκω

m dXfmVf

where 1e is the unit vector in the 1X direction, )( 3)2,( 0 XωκmV the velocity profile and

)( 3)2,( 0 Xωκmσ the stress profile of the m th Lamb mode. The superscripts of an asterisk denote

the complex conjugate. In Eqs. (15) and (16), ω2p and ω2f are the second-harmonic terms of

NLP and NLF , which are defined as

c.c.,])(exp[)(2

1

)(2exp)(2

1),,(

1*

30

0132

31LNL

XκκiX

tωXκiXtXX

llω

lω

p

pHP

c.c.])(exp[)(2

1

)(2exp)(2

1),,(

1*

30

0132

31LNL

XκκiX

tωXκiXtXX

llω

lω

f

fHF

Note that mmP is 2/h regardless of m , since the energy flux density of )( 3)2,( 0 XωκmU is

normalized. Let )( 1XRm be the ratio of the in-plane displacement amplitude of the

second-harmonic Lamb mode to the squared in-plane displacement amplitude of the fundamental

Lamb mode on the surface ( hX 3 ), i.e.,

.

)(e

)()()( 2

1),(

0

1)2,(

1

101

0

eU

eU

hA

hXAXR

ωκXκi

ωκm

mll

m

This ratio can be decomposed as

,)()( 111 XTSXXR mmm

where

In the

Lamb

The f

distan

3.2 E

symm

secon

are ca

mode

show

longit

foreg

show

frequ

from

Fig. mod

e

e above expr

b mode to the

factor (XTm

nce.

xample and D

The first-o

metric (S2) L

nd-harmonic

alculated for

e pair of 1.8

wn in the ph

tudinal displ

going experim

wn in Table 1

uency 0f . F

50 to 150 m

2 Dispersionde pair which s

Sm

)( 1m XT

ession, mS

e second-har

)1X determin

Discussion

order symme

Lamb mode

Lamb modes

the fundame

MHz S1 mo

hase velocity

acements [13

mental invest

[26]. In Fig

igures 3(a) a

mm. The ver

curves of Lasatisfies the ph

4

surf

P

ff

mm

mm

,1

( sinc mκ

contains a f

rmonic Lamb

nes the devi

tric (S1) La

with frequ

s. For this pai

ental frequen

de and 3.6 M

y-frequency d

3, 14]). This

tigations [8,

. 3, mR , S

and 3(c) sho

rtical line in

amb waves forhase-matching

6

(0

(vol

U

U

A

fκ

κ

m

mm

2

)21

lm Xκ

factor represe

b mode as dis

ation from t

amb mode w

uency 02ω

ir, mR , mS

ncy 00 ωf MHz S2 mod

diagram in

is also the p

10, 19, 20].

mS and mT

ow the result

n Fig. 3(a) sh

r an aluminumg condition

)(

)(2

1),

1)2,

0

0

e

e

h

h

ωκ

ω

l

,

κ

κ

enting the po

scussed by e

the linear de

with frequen

are selected

m and mT in

)2/( π from

de satisfies th

Fig. 2 (sym

pair of Lamb

The materia

are shown a

ts when the

hows the ex

m plate with

,

.2

,2

lm

lm

κκ

κκ

wer flux from

.g., de Lima

ependence on

cy 0ω and

as the fun

n a 2 mm-thi

m 1.6 to 2.0 M

he phase-mat

mmetric mod

modes which

al parameters

s functions o

propagation

act phase-ma

a thickness of

m the fundam

and Hamilto

on the propa

d the second

ndamental an

ick aluminum

MHz. Note th

tching condit

des with dom

h has been u

s of aluminu

of the fundam

distance is

atching frequ

f 2.0 mm and

mental

on [5].

gation

d-order

nd the

m plate

hat the

tion as

minant

used in

um are

mental

varied

uency.

d the

Tab

Alu

S

Fig.mod

mRalum

le 1 Physical p

0ρ

uminum

Steel

3 (a) Variatiode to the squa

and its comp

minum plate, a

properties of a

0 [kg/m3]

2700

7874

on of the ratioared in-plane d

ponents (b) Sand (d), (e), (f)

aluminum and

λ [GPa]

56.0

111

o of the in-pladisplacement

mS and (c) T) those for a st

7

d steel

μ [GPa]

26.5

82.1

ane displacemamplitude of

mT with the f

teel plate, resp

A [GPa]

-408

-708

ent amplitudethe fundamen

fundamental f

pectively.

B [GPa

-197

-282

e of the secondntal Lamb mo

frequency of L

a] C [G

-114

-179

d-harmonic Lode on the sur

Lamb wave fo

Pa]

4

9

amb rface

r an

Consi

passb

passb

funda

distan

since

maxim

repre

the se

mS

1.5 to

are sh

satisf

and (

which

measu

4. NU

4.1 Fi

T

Recen

(FDT

nonlin

secon

equatare ar

idering mT

band at the p

band is dete

amental frequ

nce 1X , and

mT is almo

mum value i

sented by S

econd-harmo

To examine

and mT in

o 2.3 MHz an

hown in Tabl

fies the phase

(f) that the p

h indicates t

urements is l

UMERICAL

inite Differen

To confirm t

ntly, Matsuda

TD) scheme

nearity. Ass

nd-order diff

tions. Nodes rranged in th

as a function

phase-matchi

ermined by

uency is wit

d its fundam

ost constant

s determined

mS given in

nic Lamb mo

e the influen

a 2 mm-thick

nd shown in

le 1 [26]. No

e-matching c

assband for

that the tunin

less severe fo

L SIMULATI

nce Formulat

the results of

a and Biwa [

[28] to inco

suming two

ferential equ

of the partiche grid for th

Fig.

n of the fund

ing frequenc

the propaga

thin the pass

mental freque

in the passb

d not only by

Eq. (21), i.e

ode.

nce of the dif

k steel plate

Fig. 3 (d), (e

te that the m

ondition in a

a steel plate

ng of the fu

or steel.

ION

tion for Nonl

f the perturb

[27] have ext

orporate the

-dimensional

ation in Eq.

cle displacemhe nonlinear s

4 The FDTD g

8

amental freq

cy 1.8 MHz

ation distanc

sband, mR

ncy depende

band. Theref

y the phase-m

e., the power

fference of m

are also calc

e) and (f), res

mode pair of 1

a steel plate w

is about tw

undamental f

linear Elastic

bation analys

tended the el

stress-strain

l plane-strai

. (1) is rewr

ments, the velsimulations a

grid for the no

quency, it acts

as shown in

ce of Lamb

grows in pr

ence is mainl

fore, the freq

matching con

r flux from th

material on t

culated for th

spectively. T

1.93 MHz S1

with 2mm thi

ice as wide

frequency for

c Media

is, we also c

lastodynamic

n nonlinearit

in motion

ritten into a

locities and tas shown in F

onlinear simula

s as a bandpa

n Fig. 3(c).

waves. The

oportion wit

ly determine

quency for w

dition but als

he fundamen

the harmonic

he fundament

he material p

mode and 3

ickness. We s

as that for an

r cumulative

conduct num

c finite-differ

ty as well a

in the 1X

set of first

the first PiolaFig. 4 where

ation

ass filter with

The width

erefore, whe

th the propa

ed by that of

which mR h

so by other f

ntal Lamb mo

c generation,

tal frequency

parameters o

3.86 MHz S2

see from Fig

an aluminum

e second-har

merical simula

rence time-d

as the kinem

3X plane

t-order differ

a-Kirchoff ste dΔ denot

h a flat

of the

en the

gation

f mS ,

has the

factors

ode to

, mR ,

y from

f steel

mode

g. 3 (c)

plate,

monic

ations.

omain

matical

e, the

rential

tresses tes the

9

grid size. The displacements and the stresses ( 1U , 3U , 11P , 33P , 13P , 31P ) are updated when

the time step index is integer and the velocities ( 1V , 3V ) are updated when it is half-integer. The

updated displacements are given as follows,

),,2/1(Δ),2/1(),2/1( 2/11

111 jitVjiUjiU nnn

),2/1,(Δ)2/1,()2/1,( 2/13

133 jitVjiUjiU nnn

where i and j are the indices of a grid point in the 1X and 3X direction, respectively. The

superscript index n denotes the time step and tΔ is its increment. The components of the first

Piola-Kirchhoff stress tensor are given by

),,(),(

),(),(

),(),(),(2

),(

),(),(),(

1,33,14

2

1,3

2

3,13

3,33,31,12

2

1,11

3,3131,11111

jiUjiUd

jiUjiUd

jiUjiUjiUd

jiUd

jiUcjiUcjiP

nn

nn

nnn

n

nnn

),,(),(

),(),(

),(),(2),(

),(

),(),(),(

1,33,14

2

1,3

2

3,13

1,13,31,12

2

3,31

1,1133,31133

jiUjiUd

jiUjiUd

jiUjiUjiUd

jiUd

jiUcjiUcjiP

nn

nn

nnn

n

nnn

,)21,21()21,21(2

)21,21()21,21(

)21,21()21,21()21,21(

1,343,13

3,31,1

1,33,15513

jiUdjiUd

jiUjiU

jiUjiUcjiP

nn

nn

nnn

,)21,21()21,21(2

)21,21()21,21(

)21,21()21,21()21,21(

3,141,33

3,31,1

1,33,15531

jiUdjiUd

jiUjiU

jiUjiUcjiP

nn

nn

nnn

where the subscripts following a comma denote the partial derivatives with respect to the material

coordinate. The coefficients 11c , 13c , 55c , 1d , 2d , 3d and 4d are the linear combinations of

the second- and third-order elastic constants of isotropic media defined as follows,

The e

Expli

Piola

prese

altern

index

zero-

4.2 N

S

the th

( 3X

mater

and

T

tracti

where

equation of m

(2/11 iV n

,(2/13

iV n

icit expressio

-Kirchhoff s

nted here (th

nately calcula

x is integer or

stress formul

Numerical An

Specification

hickness of t

)h are a

rial propertie

dΔ = 50 µm

The fundame

ons given as

e

motion gives t

),21 Vj

)21 Vj

ons for the sp

stress tensor

he details are

ate the displa

r half-integer

lation [28].

nalysis of Non

n of the mode

the plate are

assumed trac

es are listed i

m, respectively

ental Lamb w

,0

,0

313

311

XP

XP

Fig. 5 The c

2

12

12

1

32

3

2

4

3

2

1

11

μd

μλd

Bλd

λd

μλc

the velocities

21(2/11 iV n

1,(2/13 jiV n

patial gradien

in Eqs. (30

e described b

acements/stre

r, respectivel

nlinear Lamb

el used for th

e 400 mm an

ction-free. Th

in Table 1. T

y.

wave is excite

(,

(,

03

03

tWAt

tWAt

configuration

10

.

2

1

4

1

,

33

, 13

BA

Aμ

CB

BAμ

λcμ

s of the next

Δ),2 1

0

Pρ

tj

Δ)21 3

0

Pρ

t

nts of the disp

0) and (31)

by Matsuda

esses and the

ly. The stress

b Wave Propa

he numerical

nd 2 mm, re

he plate is a

The time step

ed from the l

Re)

Re)),(

13

),(11

0

0

σt

σtωκ

ωκ

l

l

of the model f

,

,

,, 55

B

CB

μc

time step,

,21(1,11 jiP n

21,(1,31 jiP n

placements in

are lengthy,

and Biwa [2

e velocities ac

s-free bounda

agation

analysis is s

espectively.

assumed to b

p increment a

left end ( 1X

exp()(

exp()(

3)

3)

X

X

for numerical

,

() 3,13 iPj n

,()2 3,33 iP n

n Eqs. (25) to

, and hence

27]). Using E

ccording to w

aries are inco

hown in Fig.

The upper a

be made of

and grid size

= 0) of the p

,)2

,)2

0

0

tifπ

tifπ

calculation

,),21 j

.)21, j

o (28) and th

they will n

Eqs. (23)-(31

when the tim

orporated usin

. 5. The leng

and lower su

aluminum a

e are tΔ = 3

plate, by pres

(29)

(30)

(31)

he first

not be

1), we

me step

ng the

th and

urfaces

and its

.15 ns

cribed

where,(

13σωκl

6 illu

0(11P

and a

steady

frequ

funda

funda

displa

upper

excita

in-pla

windoKFU1

where

and

e 0t = 14 [

)( 3)0 Xω

are

ustrates the

),0,0 t with r

achieves a c

y-state mono

uency 0f fr

amental Lam

amental Lam

acement amp

r surface of t

ation point.

ane displace

ow functionfK , is cal

e K and f),( 1, tXG

tX

(W

s] and 1t =

given as the

input wavef

respect to t

onstant leve

ochromatic ex

rom 1.6 MH

mb mode am

mb mode is e

plitude 1.0 nm

the plate, we

These wave

ment (1 XU

n and analylculated as

),( fKFU

f are the wa

is a Gaussia

(,G tX

F

,1

exp)t

= 4.7 [s] in

S1 mode str

form 11 0,0(P

. The amplit

l, which ena

xcitation. The

Hz to 2.0 M

mplitude 0A

qual to that

m. Collecting

e obtain the t

forms can b),1 tX . This

zed by the

G

X

avenumber (i

an window fu

exp),( 1 tX

Fig. 6 Time do

11

,

2

1

0

t

tt

n this study.

ress profiles

max11/),0 Pt ,

tude of the p

ables the sim

e simulation

MHz in step

is set so th

of the longit

g the comput

time domain

be considered

distribution

two-dimens

(),( 1,XUtX

tX

inverse of w

unction:

p 1

ξ

XX

main wavefor

0

0t

t

. The thickn

by the linear

where m11P

rescribed tra

mulation of t

is performed

s of 0.01 M

hat the energ

tudinal wave

ted in-plane d

n waveforms

d as a spatia

is windowe

sional Fouri

e), 21

1tX KXiπ

ave length) a

2

τ

ttX

m of excitatio

,

,0

t

t

ess profiles

theory of La

max is the m

ction rises gr

the response

d by increasin

MHz. In eac

gy flux dens

e with freque

displacement

at different d

al-temporal d

ed by a tem

ier transform

,1dtdXft

and the frequ

.2

t

n

)( 3),(

110 Xσ ωκl

amb waves. F

maximum val

gradually with

e of the plate

ng the fundam

ch simulation

sity of the e

ency 1.8 MH

t waveforms

distances fro

distribution

mporal and s

m. The tran

uency, respec

) and

Figure

lue of

h time

e to a

mental

n, the

xcited

Hz and

at the

om the

of the

spatial

nsform

tively,

The w

spatia

wave

4.3 R

Ffrequ

[mm]

Fig. 7

windo

theory

and th

the ra

and s

funda

the l

propo

MHz,

MHz

can b

sense

I

frequ

curve

window func

al position o

.

Results and di

Figure 7 shouency is 1.8 M

], and the spa

7 are not shar

ow function.

y of Lamb w

hat of the fun

atio of the se

shown in Fig

amental frequ

lines are c

ortionally wi

, but cumulat

mR can be

be generated

e.

In Fig. 9 th

uency when t

es denote the

Fig. 7 f

tion has four

of the windo

iscussion

ows the resuMHz. The w

atial position

rp but show

The solid an

waves. The a

ndamental m

econd harmon

. 8 as functio

uencies. The

alculated by

th propagatio

tive growth c

e even higher

cumulatively

he plotted sy

the spatial po

e results of t

K distributi

r parameters

w function

ult of the twwindow param

of the windo

some spread

nd dashed lin

amplitude of

mode (S1 mod

nic amplitude

ons of the sp

symbols in F

y the pertu

on distance w

can also be se

r at finite dis

y when the p

ymbols deno

osition of th

the perturbat

ion of log10

12

which determ

X , corresp

wo-dimensionmeters are tow function

d because of t

nes in Fig. 7

the second-h

de, 1.8 MHz)

e and the squ

patial position

Fig. 8 show

urbation ana

when the fre

een at the ne

stances. This

phase-matchi

ote mR as f

he window fu

tion analysis

),(1

fKFU

mine their sh

onding to th

nal Fourier tt = 180 [s

X is 100 m

the finite tem

7 denote the

harmonic Lam

) can be obta

uared fundam

n of the wind

the results o

alysis. The

equency of fu

eighboring fre

s shows that

ng condition

functions of

unction Xs shown in F

at the propag

ape and posi

he propagatio

transform wh], τ = 3.3

mm. It is note

mporal and sp

dispersion cu

mb wave (S2

ained from Fi

mental wave a

dow function

f the FDTD

amplitude

undamental L

equencies. In

second-harm

n is satisfied

the fundame

is 50, 100,

Fig. 3(a), and

gation distanc

ition, includi

on distance

hen the exci[s] and ξ

ed that the pe

patial widths

urves of the

2 mode, 3.6

ig. 7. We cal

amplitude as

n X , for dif

simulations,

ratio mR

Lamb mode

n particular, f

monic Lamb

in an approx

mental Lamb

and 150 mm

d the vertica

ce of 100 mm

ng the

of the

itation = 6.7

eaks in

of the

linear

MHz)

lculate

s mR ,

fferent

while

grows

is 1.8

for 1.9

waves

ximate

mode

m. The

al line

show

show

proba

pertur

when

P

gener

degra

funda

in pra

not ex

true p

ws the phase-m

w fair agreeme

ably due to th

rbation analy

n the fundame

Practically, it

rates a high

adation. This

amental frequ

actical situati

xactly known

phase-matchi

F

Fig. 9 Vari

matching freq

ent with thos

he numerical

ysis, the resu

ental frequen

t is an impor

h mR to as

s study has

uency is not

ions. For exa

n and the fun

ing frequenc

Fig. 8 Variation

iation of the a

quency. In Fi

e of the pertu

l dispersion

ults of FDTD

ncy is close bu

rtant issue to

ssess the ma

revealed th

exactly equa

ample, even i

ndamental fre

y, nearly cum

n of the amplit

mplitude ratio

13

igs. 8 and 9,

urbation anal

inherent in t

D simulations

ut not equal

select the fu

aterial nonli

hat mR can

al to the phas

f the acoustic

equency chos

mulative gro

tude ratio mR

o mR with fu

the numerica

lysis, althoug

the FDTD m

s also show t

to the phase-

undamental fr

nearity sens

n become re

se-matching f

c properties o

sen in the me

owth of seco

m with propa

undamental fr

al results by

gh there are s

ethod. As sh

hat mR rea

-matching fre

requency of L

sitively in th

elatively high

frequency. Th

or the thickne

asurement is

nd harmonic

gation distanc

requency of La

the FDTD m

some discrep

hown above b

aches its max

equency.

Lamb mode

he early sta

h even whe

This is a usefu

ess of the pla

s different fro

c can be exp

ce

amb waves

method

ancies

by the

ximum

which

age of

en the

ul fact

ate are

om the

pected.

14

This fact shows robustness of the material evaluation method using second harmonics in Lamb

waves. It should be noted, however, that the linear dependence of the second-harmonic amplitude

on the propagation distance can be violated when the fundamental frequency is off the phase

matching: the frequency bandwidth in which the linear dependence holds is determined by the

propagation distance as shown in Fig. 3(c) for aluminum and Fig. 3 (f) for steel. This needs to be

accounted for when material nonlinearity is to be quantitatively evaluated from the measurement.

It is finally commented that in contrast to the monochromatic excitation simulated here,

practical measurements are more often performed using a tone-burst excitation of the fundamental

Lamb wave which has a finite frequency bandwidth. As far as weak nonlinearity holds, it is

expected that the dependence of the observed second-harmonic amplitude on the center frequency

of the tone-burst is given by a (kind of) average of the results shown here for monochromatic

excitation over the frequency bandwidth. As a result, peaks of the second-harmonic amplitude for

varying center frequency will be less sharp than those shown here, and the null points of the

second-harmonic amplitude (as seen at 1.64 MHz and 1.96 MHz for 150 mm in Fig. 9) will

disappear.

5. CONCLUSION

An analysis of the fundamental frequency dependence of the second-harmonic generation in

Lamb waves has been shown in this paper. The ratio of the amplitude of the second-harmonic

Lamb mode to the squared amplitude of the fundamental Lamb mode has been calculated by the

perturbation analysis and modal decomposition. The second-harmonic amplitude can grow

cumulatively in a certain range of fundamental frequency when the propagation length is finite.

The analysis reveals that the frequency for which the second harmonic Lamb mode reaches

maximum is close but not equal to the phase-matching frequency in a precise sense. This feature

has been confirmed numerically using the finite-difference time-domain method incorporating the

material and geometrical nonlinearities. The fact that the second-harmonic amplitude becomes

high in a certain fundamental frequency range proves robustness of the material evaluation method

using second harmonics in Lamb waves.

6. ACKNOWLEDGEMENTS

This work has been supported by JSPS KAKENHI Grant Numbers 24·2517 and 25289005.

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