Title Frequency Dependence of Second-Harmonic...
Transcript of Title Frequency Dependence of Second-Harmonic...
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 [email protected]
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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