Transient 3D elastodynamic field in an embedded multilayered anisotropic plate

(1)

is an open access repository that collects the work of Arts et Métiers Institute of

Technology researchers and makes it freely available over the web where possible.

This is an author-deposited version published in:

https://sam.ensam.eu

Handle ID: .

http://hdl.handle.net/10985/10734

To cite this version :

Pierric MORA, Eric DUCASSE, Marc DESCHAMPS - Transient 3D elastodynamic field in an

embedded multilayered anisotropic plate - Ultrasonics - Vol. 69, p.106-115 - 2016

Any correspondence concerning this service should be sent to the repository

Administrator :

[email protected]

(2)

Transient 3D elastodynami eld in

an embedded multilayered anisotropi plate

Pierri Mora 1 pierri .morau-bordeaux.fr Eri Du asse 1,3,4 eri .du asseu-bordeaux.fr Mar Des hamps 1,2 mar .des hampsu-bordeaux.fr 1

Univ. Bordeaux,

I

2 M´AP y

, UMR5295, F-33400Talen e,Fran e. 2

CNRS,

I

2 M

´AP y

,UMR 5295,F-33400Talen e,Fran e. 3

Arts etMetiersParisTe h,

I

2 M

´AP y

,UMR 5295,F-33400Talen e, Fran e. 4

Correspondingauthor. Tel. +33(0)540003138. Fax+33(0)540006964.

Abstra t

The aimof thispaperis tostudy the ultrasoni response toatransient sour ethat radiatesultrasoni waves in

a3D embeddedmultilayeredanisotropi anddissipativeplate. Thesour e an be insidethe plateoroutside,in

auidloadingthe plateforexample. Inthe ontextofNon-Destru tiveTesting appliedto ompositematerials,

our goal is to reate a robust algorithm to al ulate ultrasoni eld, irrespe tive of the sour e and re eiver

positions.

The prin iple of the method des ribed inthis paper is well-established. This methodis based ontime analysis

using the Lapla e transform. In the present work, it has been ustomized for omputing ultrasoni sour e

intera tions withmultilayereddissipativeanisotropi plates. Theeldsare transformedinthe2D Fourier

wave-ve tor domain for the spa e variables related to the plate surfa e, and they are expressed in the partial-wave

basis. Surprisingly, this method has been very little used in the ultrasoni ommunity, while it isa useful tool

whi h omplements the mu h used te hnique based on generalized Lamb wave de omposition. By avoiding

mode analysis whi h an be problemati in some ases exa t numeri al al ulations (i.e., approximations

by trun ating innite series that may be poorly onvergent are not needed) an be made in a relatively short

time for immersed plates and vis oelasti layers. Even for 3D ases, numeri al osts are relatively low. Spe ial

attention is given to separate up- and down-going waves, whi h is a simple matter when using the Lapla e

transform.

Numeri al results show the ee tiveness of this method. Three examples are presented here toinvestigate the

quality of the modeland the robustness of the algorithm: rst, a omparison of experiment and simulation for

a monolayer arbon-epoxy plate,where the dira tedeld is due toasour e lo ated onthe rst freesurfa e of

the sample,forboth dissipativeandnon-dissipative ases; se ond, thebasi ongurationof analuminum plate

immersedinwaterhasbeen hosentostudywavepropagationinZGV(ZeroGroupVelo ity) onditions;nally,

a2Dplate onsistingof8sta ked arbon-epoxylayers immersedinwater istreated, withasour elo atedinside

(3)

Highlights

Transientresponse omputation of a3D embedded multilayered anisotropi plate.

Ultrasoni eld al ulation,irrespe tive of the sour eand re eiver positions.

Standard numeri al integraltransform may provide afast and e ient tool.

Up- and down-going waves are easily sele ted when using the Lapla e transform.

Keywords

Embedded multilayered plate ; Transient ultrasoni response ; Partial-wave expansion ; Integral transform

domain ;Numeri al inverse Lapla e transform

PACS 43.20.Bi ;43.20.Mv

1 Introdu tion

The study of dira tion of a ousti waves in laminated plates has interested the ommunities of geophysi s,

underwater a ousti s and non-destru tive testingfor several de ades.

Numerous pra ti al approa hes exist today to al ulate the response of these media to dierent sour es. The

hoi e ofmethodused isgoverned bythe situationunder onsideration. Forinstan e, the aimmaybenear-eld

al ulationorfar-eld al ulation,theresponsesmaybeneeded inthetimeorthefrequen y domain,the guides

may or may not be loaded, et . Expanding the solution in terms of normal modes in the frequen y domain is

one of the best-known methods. The modes orrespond to generalized Lamb waves for laminated plates, and

an be obtained numeri allyindierent ways, by sear hing for the rootsofthe determinantofa matrix[14℄, or

by obtainingthe eigenvalues of anite-elementmatrix (e.g., [13, 8℄).

This modalmethod is well-suited to deal with al ulations in the far-eld domain for guides in va uum. This

is be ause in the far eld the modes are well-built, and only a limited number of them, i.e., the propagative

modes, ontribute, whi h makes modal theory very e ient. For plates in va uum, the solution an also be

expanded in terms of modes in the time domain, after performing a Fourier transform with respe t to the

horizontal oordinates (inthe plane of the plate)[12, 7℄,assumingthat the sour e isbounded and bandlimited

in horizontal wavenumber, a ording to the Nyquist-Shannon sampling theorem. This se ond methodis more

suitable than the rst one for 3D al ulationsin anisotropi media be ause itdoes not make use of anangular

integration variable,nor is the dire tionof propagationof the modes required [26℄.

However, for loaded plates (i.e., plates immersed or embedded in an unbounded medium), modes do not

onstitute a omplete basis, whi h makes modal expansion theoreti ally more deli ate, if not impossible.

Furthermore, it is still a hallenge to build a robust and a urate algorithmto obtain the modes numeri ally.

Indeed, exa t eigenvalue statements have been proposed only in some restri ted ases [9℄. For more general

situations, PML [4℄ are sometimes hosen for modeling the unbounded layers, but are well-known to be

problemati inthe aseof ba kward propagativemodesand mayalsoleadtoinstabilitiesfor anisotropi media.

So, inthis ase this modalte hnique must beused with some aution.

To avoid di ulties due to radiation into half-spa e(s), dierent approa hes have been developed. Park and

Kausel [17, 18℄ have shown that, for loaded guides, it is possible to get an exa t solution from the modal

basis obtained for guides in va uum. The inuen e of the two semi-innite media is taken into a ount by

(4)

determinedby solvingaVolterraintegralequation. However, thismethod,the so- alledSubstru tureMethod,

exhibits poor onvergen e and problems of instability [17, 18℄.

An alternative to the modal theory onsists in onserving the initialintegral instead of expressing it as a sum

of modes by using the residue theorem. The problem is then to al ulatethis integral orre tly, knowing that

sampling the integration path, whi h is lose to the poles, an produ e numeri al errors. To this end, in the

ase ofasta k of3Disotropi platelayers, in1965,Phinney[19℄ proposedtodeformthe integration pathofthe

Fourierintegralonfrequen y inthe omplexplanebyaddinganarbitraryimaginarypart tothe frequen y. The

hoi e of the arbitrary onstant has been studied by Kausel and Roësset [11℄ inorder to in rease the a ura y

of the numeri al inverse transform involved to ome ba k to the time domain. This method an be found

in the literature under the name of Exponential Window Method. Although it is never stated expli itly, all

the numeri alinversions relatedto the Exponential Window Methodgive ausal signals and orrespond tothe

inverse Lapla e transform dened by the Bromwi h ontour. The same idea has been used to deal with wave

dira tion in 3D isotropi dissipative (or non-dissipative) plates [28℄. This work has been extended to model

ultrasound generation by laser in 2D dissipative anisotropi plates by Audoin and Guilbaud[1℄.

Curiously, to our knowledge, this ausal non-modal te hnique is used relatively infrequently by the me hani s

ommunity, even though it is able to solve omplex ases while insuring a perfe t ontrol and ompetitive

numeri al ost towards the modal methods. This is probably be ause it is a purely numeri al te hnique, for

whi h the physi al meaning ontained in modaltheory is lost. Ifa pre ise physi al interpretation of dira ted

wavefronts is needed, both modaland non-modalapproa hes are then ne essary.

Inthis paperweproposetostudy theultrasoni responsetoatransientsour ethat radiatesultrasoni wavesin

a3D embeddedmultilayeredanisotropi anddissipativeplate. Thesour e an be insidethe plateoroutside,in

auid loadingthe platefor example. In someway, the goalof this paperistogive tothiswell-established

non-modal method a larger diusion among the ultrasoni ommunity. The examples presented here are hosen to

overabroadrangeofsituationsofNon-Destru tive Testing (NDT)ofplates,withaspe ialfo uson omposite

plates, onsidering guided wave as well as bulk wave regimes. The formalism may be readily applied to write

a robust and versatile algorithm to al ulate ultrasoni eld, irrespe tive of the sour e and re eiver positions.

The elds are transformed in the Lapla e domain for the time variable and in the 2D Fourier wave-ve tor

domainfor the spa evariablesrelatedtothe horizontalplate surfa e, assumingthatthe sour e isbounded and

bandlimited in horizontal wavenumber, a ording to the Nyquist-Shannon sampling theorem. Working in the

Lapla edomaingivesa esstoallthegeneraltheoremsasso iatedtothistransformationandspe i allyrelated

to ausal signals. Beyond the spe i interest of this paper, this an be useful for building solutionsto many

t

notation Lapla e-domain

s

notation

Physi al spa e

x, z

px, z, tq

u

px, z, sq

U

Horizontalwave ve tor

k

Verti al position

z

pk, z, tq

u

˜

_{pk, z, sq}

U

˜

Horizontalwave ve tor

k

Verti al wavenumber

k

z

pk, k

z

, t

q

u

ˆ

pk, k

z

, s

q

U

ˆ

Table1: Notations

2 Notations

The ve tor

x

“ px, yq

willdenote spatial oordinates inthe

xy

-plane. The asso iatedplanar wave-ve tor inthe Fourier spa ewillbedenoted by

k

“ pk

x

, k

y

q

. In thetime domain,alleldsare denoted with lower ase letters,

whereas in the Lapla e domain apital letters are used. Symbols are added to indi ate a Fourier transform

over the spa e variables. These onventions are summarized in Table 1. The elds

u

,

σ

z

, η, f

β

, σ

β

that are introdu edlater followthe same rules: they willbedenoted

˜

U, ˜

Σ

z

, ˜

H, ˜

F

β

, ˜

Σ

β

in the

pk, z, sq

dual spa e. The

dependen e of these elds along the spa e and spe tralvariableswillbemade expli it onlywhen ne essary.

The me hani al properties

ρ

β

and

c

β

ijkm

are onstant in ea h layer. Noti e that the

z

axis is taken positive

downwards, and that therefore an upgoing (resp. downgoing) wave propagates towards de reasing (resp.

in reasing)

z

.

3 Equations and ba kground

Let us onsider a multilayered mediummade of a plate system onsisting of a number

N

of perfe t at layers

of normal

n

along

z

-axis, sta ked together. The layers are labeled

β

. The interfa e between layers

β

and

β

`1

, also labeled

β

, is lo ated at position

z

“ z

β

, as illustrated in Fig. 1. Ea h layer is an anisotropi solid, with a giventhi kness

h

β

“ z

β

´ z

β´1

. The areasaboveand belowthis platesystem aneitherbeva uaorsemi-innite

half-spa esof solid (oruid) media. The plate isassumed to beinnite inthe

xy

-planeThis stratied medium

is submitted to external for es that an be lo ated anywhere. The rst interfa e (

β

“ 0

) is always lo ated at

z

0 “ 0

.

(6)

3.1 Basi equations in the physi al spa e

With our notations, let us rst state the wave equations in ea h layer. Combining Newton's se ond law, the

ausality prin iple and Hooke's law, the displa ement eld

u

px, z, tq

is given, at any time

t

and any lo ation

px, zq

, by the following system expressed inthe medium

β

:

$

’

&

’

%

ρ

β

B

2 t

u

px, z, tq ´ p▽

β

˛ ▽q upx, z, tq “ f

β

px, z, tq, for t ą 0,

u

_{px, z, tq “ 0,}

for

t

_{ă 0,}

σ

_z

_{px, z, tq “ pn}

β

_{˛ ▽q upx, z, tq,}

(1)

wherethe quantity

σ

z

px, z, tq

expresses the stress inthe

z

-dire tion(normaltothe interfa es)and

ρ

β

standsfor

the mass density. The bilinear produ t

β

˛

has been dened in [6℄ by a three-by-three matrix

pa

β

˛ bq

su h that

pa

β

˛ bq

im

“ c

β

ijkm

a

j

b

k

, with the Einstein summation onvention. The above equations depend on ea h layer through the values of the elasti onstants

c

β

ijkm

, i.e. through the operator

β

˛

. The eld

f

β

px, z, tq

denotes the for e perunit volume exerted by the partof the sour eslo atedin layer

β

.

Continuity equations atthe interfa e

β

are given by the followingsystem:

ˆ

u

_{px, z}

_β

`

, t

_q

σ

_z

_{px, z}

_β

`

, t

_q

˙

´

ˆ

u

_{px, z}

_β

´

, t

_q

σ

_z

_{px, z}

_β

´

, t

_q

˙

“

ˆ

0 σ

_β

_{px, tq}

˙

,

(2) where

z

´

β

and

z

`

β

indi ate the fa t that the eld under onsideration is al ulated in layers

β

´1

and

β

,

respe tively. The interfa e sour e term

σ

β

px, tq

denes the normalstress jump atinterfa e

β

, and orresponds toan appliedfor eperunit surfa e if one ofthe media isa va uum. Noti e that if

σ

β

“ 0

, then Eq.(2)merely

expresses the ontinuity of displa ementand normal stress.

Forsimpli ity,the

β

indexwillbeomittedbelowandwillbereintrodu edonlywhenne essarytoavoidambiguity. Thevolumi sour esdes ribedby

f

β

px, z, tq

inEq.(1) anstraddletwoormorelayers. Insu h ase,thevolumi sour e term is onsidered aszero at ea h interfa e (

z

“ z

β

).

Eq. (1) and (2) are tobe solved using Fourier and Lapla e transforms oninvariantdimensions. In a rst time

(see se tions 3.2 and 3.3), Eq. (1) is solved separately in ea h layer ontaining a sour e term. This denes an

in ident eld inea h layer, whi h orresponds to the eld that a sour e would radiate inthis layer onsidered

tobe unbounded. Then(see se tion3.4),the ree ted eld an be obtained,whi histhe ontributionofallthe

interfa es, and is al ulated by taking intoa ount the ontinuity relationships of Eq. (2). In this arti le, the

so- alled Global matrix (e.g.,[14℄) method isadopted toperform the latter operation. The total eld isnally

obtained as the sum of the in ident and ree ted elds.

3.2 Separation of up- and down-going waves

The aim of this se tion is to emphasize that up- and down-going bulk waves, whi h are the solutions of

Christoel's equation, are easily separated inthe Lapla e domain. It may beskipped ina rst reading.

Let us apply a 3D Fourier transform to the spa e variables to Eq. (1). The spa e domain

px, zq

is then

transformed intothe spe tral domain dened by

K

“ pk, k

z

q

, where

K

stands for the wave ve tor. This yields

the following dierentialequation:

ρ

_B

_t

2 u

ˆ

_{ptq ` pK ˛ Kq ˆu ptq “ ˆf ptq .}

(3)

If

K

“ 0

, whi h orresponds to the rigid body motion, then

u

ˆ

“ ρ

´1

r

˚ ˆf

, where r denotes the ramp fun tion and theoperator

˚

isthetime onvolutionprodu t. If

K

‰ 0

,thematrix

pK ˛ Kq

isthen real-valuedsymmetri

(7)

positive-denite,and thereforeadmitsthreepositiveeigenvalues

ρ ω

2 i

withasso iatedeigenve tors

P

i

su h that:

pK ˛ Kq “ ρ P diag

1 _ďiď3

`

ω

_i

2 ˘

P

T

,

(4)

wherethe3x3polarizationmatrixissu hthat

P

“ rP

1 , P

2 , P

3 s

. Itstransposematrixisnoted

P

T

. Thesolution

is then expressed asfollows:

ˆ

u

_{ptq “}

1 ρ

P

1 diag

_ďiď3

ˆ

h

p‚q sinpω

i

‚q

ω

i

˙

P

T

_{˚ ˆf,}

(5)

wherehis theHeavisideunitstep fun tion. Letusnow perform aLapla etransform(withparameter

s

)tothe

solution(5). The Lapla etransformsof the ramp and sine fun tions are

s

´2

and

p1 ` s

2 q

´1

,respe tively. They

are holomorphi onthe half-plane

Re

psqą0

, with poles on the imaginaryaxis.

If the 6D state ve tor

ˆ

H

_“

´

_U

ˆ

T

ˆ

Σ

T

_z

¯

T

is onsidered, the solution an be expressed in the dual spa e

pK, sq

as follows:

ˆ

H

_{“ M}

´1

ˆ

0 ˆ

F

˙

,

where

M

_“

ˆ

i

pn ˛ Kq

I

ρ s

2

I

` pK ˛ Kq

O

˙

,

(6) i

“

?

´1

,I and O are the 3-by-3 identity and zero matri es, respe tively. The study of the singularities of

ˆ

H

leads to the Christoel Equation

det

pMq “ 0

. A ording to the problem

geometry,the six solutionsof this equation are to be found in terms of omplexverti al wavenumbers

k

z,i

pi “

1 . . . 6

_q

. Thus, the state ve tor dened in Eq. (6) an be easily expressed in the

pk, z, sq

domain with the notation

˜

H

_pzq

. The six omplex verti al wavenumbers are sorted by omparing their imaginary parts, su h

that:

#

Im

_pk

z,i

q ą 0, i“1, 2, 3 ;

Im

_pk

z,i

q ă 0, i“4, 5, 6 .

(7)

Sin e

Re

psq ą 0

, it is proved in Appendix A that the rst three partial waves, of verti al wavenumbers

with positive imaginary parts, orrespond to waves that propagate in the negative

z

dire tion, while the

others, of verti al wavenumbers with negative imaginary parts, orrespond to waves that propagate in the

positive

z

dire tion. This is an important point be ause a simple onsideration on the imaginary part of

k

z,i

unambiguouslyseparates the eldsintoupgoingand downgoingwaves. Thisisnotthe aseforharmoni waves,

where

Re

psq “ 0

and

Im

psq “

i

ω

, for whi h onsiderationson dire tionof energy propagationmust be added.

3.3 Radiation of a sour e in an innite medium: in ident eld

z

Figure2: Thethreeregionsofspa edenedbyavolumesour e,inwhi hthein ident eld is al ulatedrespe tively

withEq.(8),(11)or(12).

Let us rst al ulate the radiationof a given sour e lo ated in the zone dened by

z

β,min

ď z ď z

β,max

, in the

layer

β

onsidered as an innite spa e, as shown in Fig. 2. If the sour e extends beyond the interfa e

β

(or

β

_{` 1}

), the a tual sour e to be onsidered is the part of the sour e lo ated between

z

“ z

β

and

z

“ z

β`1

,

(8)

As detailedin Appendix A, the radiatedeld, represented by the state ve tor

˜

H

inc

_β

atthe verti al position

z

in the

pk, z, sq

-domain,depends onthe position of theobservationpointwith respe t tothe sour e. It isobtained

by onvoluting the Green tensor ofthe unbounded medium by the sour e term.

As a matterof fa t, three ases o ur:

‚

If the observation point is inside the sour e area (

z

min

ă z ă z

max

),

˜

H

inc

β

pzq “

6 ÿ

i“1

α

β,i

pzq ξ

β,i

,

(8)

where

ξ

β,i

isthepolarizationve torofthemodewithverti alwavenumber

k

z,β,i

,and

α

β,i

pzq

representsthe ontributionofthis mode. Forupgoingwaves(

i

“1, 2, 3

),onlythe partofthesour ebelowtheobservation point is involved:

α

β,i

pzq “

ż

z

max

z

g

_β,i

T

_{pz ´ ζq f}

β

pζq

d

ζ

pi“1, 2, 3 and z ă z

max

q ,

(9)

where the ve tor

g

β,i

is relatedto the Green tensor inthe layer

β

(see Appendix A).

Conversely, for downgoing waves (

i

“4, 5, 6

), only the part of the sour e above the observation point

ontributes:

α

β,i

pzq “

ż

z

min

g

T

_β,i

_{pz ´ ζq f}

β

pζq

d

ζ

pi“4, 5, 6 and z ą z

min

q .

(10)

‚

If the observation point is above the sour e (

z

ď z

min

), only the upgoing waves ontribute. Then, the

state ve tor takes the followingform:

˜

H

inc

_β

_{pzq “}

3 ÿ

i“1

α

β,i

exp

r´

i

k

z,β,i

pz ´ z

min

qs ξ

β,i

,

(11)

with

α

β,i

“ α

β,i

pz

min

q

.

‚

If the observation point is below the sour e

z

ě z

max

, only downgoing waves are re eived and the state

ve tor be omes:

˜

H

inc

_β

_{pzq “}

6 ÿ

i“4

α

β,i

exp

r´

i

k

z,β,i

pz ´ z

max

qs ξ

β,i

,

(12) with

α

β,i

“ α

β,i

pz

max

q

.

Of ourse, if there isno sour e inthe layer

β

,the oe ients

α

β,i

pzq

are all zero.

At this point,we onsider that allthe in ident elds are known inall layers and hara terized in the

pk, z, sq

-domain by

˜

H

inc

β

pzq

.

3.4 Elastodynami elds in a layer and interfa e onditions

Letusnowrememberthatthe mediumunder onsiderationis bounded (

z

β´1

ă z ă z

β

). Thepart ofthe sour e

lo atedin ea h layerradiates an in ident eld hara terized by the state ve tor

˜

H

inc

β

pzq

. The total eld inthe

layer is the superposition of this in ident eld with the ree ted eld, resulting from the dira tion at ea h

interfa e. Consequently, the ree ted eld is ompletely hara terizedby six oe ients

a

β,i

and the total eld is expressed asfollows:

˜

H

β

pzq “ ˜

H

inc

β

pzq `

6 ÿ

i“1

(9)

where the verti al positions

z

β,i

an be arbitrarily hosen[23℄,as detailedbelow.

In the upper layer, the ree ted eld ontains only upgoingwaves:

˜

H

0 pzq “ ˜

H

inc

0 pzq `

3 ÿ

i“1

a

0 _,i

exp

r´

i

k

z,0,i

pz ´ z

0 ,i

qs ξ

0 ,i

,

z

ă 0 ,

(14)

while in the lower layer, the ree ted eld ontains only downgoingwaves:

˜

H

N`1

pzq “ ˜

H

inc

N

`1

pzq `

6 ÿ

i“4

a

N`11,i

exp

r´

i

k

z,N

`1,i

pz ´ z

N`1,i

qs ξ

N

`1,i

,

z

ą z

N

.

(15)

The

6N

`6

oe ients

a

β,i

willbe determined by rewriting the boundary onditions (2) in the

pk, z, sq

-domain

at the

N

`1

interfa es (6 equationsat ea hinterfa e):

˜

H

β`1

pz

β

q ´ ˜

H

β

pz

β

q “

ˆ

0 ˜

Σ

β

˙

,

(16)

where the term

˜

Σ

β

orresponds to the interfa e sour e term

σ

β

px, tq

in this transformed domain.

The arbitraryverti alpositions

z

β,i

dened in Eq.(13) have been hosensu h that:

z

β,i

“ z

β

for upgoingwaves (

i

“1, 2, 3

)and

z

β,i

“ z

β´1

fordowngoingwaves(

i

“4, 5, 6

). Inagreementwiththedenition[Eq.(7)℄,this hoi e

leads to omputations only of de reasing exponentials in the layer under onsideration. As a onsequen e,

omputations are numeri ally stable irrespe tive of the frequen y-thi kness produ ts. The in ident wave eld

radiated by the part of the sour e lo ated in layer

β

is expressed by Eq. (11) atinterfa e

β

´1

(upgoing waves

over the layer) and by Eq. (12) atinterfa e

β

(downgoingwaves under the layer).

The methodpresentedhereisknown astheGlobalmatrix approa h(e.g.,[14℄). However, itismostoftenstated

intheharmoni domain,whereasweareworkingintheLapla edomain. Notethatonlythe aseofelasti layers

embedded between two semi-innite solids is presented above for larity (giving an invertible linear system of

6N

_`6

equationswith

6N

`6

unknowns

a

β,i

). Othertypesoflayerorhalf-spa e(uid,va uum,et .) givesimilar equations.

3.5 Expression of the solution in the physi al domain

Let usrst ome ba k tothe timedomain. From a numeri alpoint of view, this isthe most deli atepro edure

of this method, sin e an inverse Lapla e transform is involved. As shown above,

˜

H

β

pzq

is holomorphi in the omplex half-plane

Re

psq ą 0

. Consequently, allitssingularities,i.e., poles, bran h pointsand bran h uts,are in the leftpart of the omplexplane, in ludingthe imaginaryaxis. Purely imaginarypoles

s

“

i

ω

0

orrespond topure guidedmodesof angularfrequen y

ω

0

. Poles with anegativereal part are asso iatedtotransient leaky

guided modes. In fa t,although the horizontal wave ve tor

k

is real, the asso iated slowness ve tor, su h that

S

_“

i

s

´1

_k

, is omplexsin e the modalfrequen y is omplex,i.e.,

Re

psq ‰ 0

(e.g.,[20℄). In su h a des ription,

wave leakage in the upperand lowerinnite mediaof this transient guidedwave is des ribed by the imaginary

part of the slownessve tor, whi h ome from transientee ts, due tothe real part of

s

.

Under these onsiderations, the inverse Lapla e transform an be al ulated, without di ulty, using the

Bromwi h-Mellin formula:

r

η

_β

pz, tq “

exp

_{pγ tq}

2 π

ż

`8

´8

exp

_p

i

ω t

q ˜

H

β

pzq

d

ω ,

z

β´1

ă z ă z

β

,

(17)

wheretheparameter

γ

isanarbitrarypositive onstantandwherethefun tion

˜

H

β

pzq

is al ulatedfor

s

“ γ`

i

ω

.

(10)

multiplied by a growing exponential fun tion. In the literature this numeri al Lapla e inversion is sometimes

referred as the Exponential Window Method (e.g., [11℄). The prin iple is to al ulate the transient response

in the dis retized time-interval

r0, t

max

s

. The use of dis reteFouriertransform results in a periodizationof the signals(period

t

max

). Following[11℄,theusualwaytodene

γ

istotake

γ

“ m logp10q t

´1

max

,whi h orrespondsto

periodizationee tsless than

10 ´m

. The greaterthe parameter,the less theperiodizationee ts. Nevertheless,

there is a ounterpart in in reasing

m

due to the exponential growing fa tor

exp

pγ tq

in Eq. (17). A good

ompromise is obtained by taking

m

between

2

and

5

.

At this stage, letus omeba k to the spatial domain,by applying the following2D inverse Fouriertransform:

η

_β

px, z, tq “

1 4π

2 ĳ

R

2 exp

_p´

i

k

¨ xq r

η

β

pz, tq

d

k

x

d

k

y

,

z

β´1

ă z ă z

β

.

(18)

A 2D inverse Fourier transform, omputed by an FFT algorithm, is then used to do this transformation,

a ording to the Nyquist-Shannon sampling theorem. As long as this inverse transform is numeri al, the

problemis spatiallyperiodi and the solutionis validonlyaslong as the fastest wave front, thatpropagates at

velo ity

c

θ

max

inaxed dire tion,hasnot rea hed theedgesofthe spatialdomain. Therefore, the omputational

eort is proportional to

k

2 max

ˆt

3 max

(

k

max

ˆt

2 max

for 2D omputations), where

k

max

denotes the maximum value

of the horizontalwave numbers of the sour e-term,assumed tobebandlimited inwavenumber.

3.6 Vis oelasti layers

To take the vis oelasti behavior of ea h layer into a ount, a Kelvin-Voigt dissipative model an be used [2℄.

As a result, the bilinear produ t

β

˛

, introdu ed in Eq. (1), is repla ed by

β

˛ `

β

˛B

t

in the time domain. In the

Lapla e domain,this hangingis notedas follows:

β

˛ ÞÑ

˛ ` s

β

˛ .

β

(19)

The new bilinear operator

β

˛

denes a 3-by-3 matrix

pa

β

˛ bq

, su h that

pa

β

˛ bq

im

“ η

_ijkm

β

a

j

b

k

, where the

onstants

η

β

ijkm

onstitute the vis osity tensor. This model is ausal. In addition, in the spe trum domain,

i.e., waveve tor/frequen y domain, it leads to a vis ous damping of waves proportional to the square of

the frequen y [2℄. It des ribes orre tly the behavior of polymers in the usual frequen y range of NDT and

NDE appli ations. However, it is not a urate for modeling omposite materials. Indeed, it has been shown

experimentally[10℄ that wave damping isproportionalto frequen y inthe domainof interest.

In the Lapla e domain,su h behavior an be des ribed by the following simplisti hange:

β

˛ ÞÑ

#

_β

˛ `

i

sign

pImpsqq

β

˛ ,

Im

psq ‰ 0 ;

β

˛ ,

Im

_{psq “ 0 .}

(20)

Note that the simplisti model (20) annot be orre t be ause it does not lead to a holomorphi expression

with respe t tothe omplexvariable

s

and doesnot orrespondtoa ausal model, ontrarytothe Kelvin-Voigt

dissipativemodel. More rened models an be used but thesewillbedis ussed infuture arti les. Nevertheless,

the simplisti model gives a eptable results for the numeri al ases presented here. The onsequen es of su h

non-physi al behavioron the waveform omputationswillbe seen below (see Se tion4.3).

Of ourseinthis ase,the onstants

η

β

ijkm

donot havethe same rheologi almeaning, orthesame unit, asthose

dened inEq. (19). However, the same variables willbe used todene the operator

β

˛

.

(11)

4 Numeri al results

This se tion reports some numeri al results based on the numeri al s hemes des ribed above. Three typi al

plates are analyzed. First, a plate in va uum onsisting of a single layer of a unidire tional ber omposite is

onsidered. Se ond, the ee ts of negative propagationin an aluminum plate immersed in water are analyzed.

Third, the ase of a stru ture made of 8 unidire tional ber omposite layers sta ked together and immersed

in water is presented. For all the examples, the omputation time will be given for a C++ language and the

omputer will be a

1.66 ˆ2 GHz

64-bit dual ore pro essor with a

2 Go

RAM, whi h is a mid-performan e

personal omputer. Usinga newer omputer, omputationtimes an easilybe divided by ten.

4.1 Monolayer plate of arbon-epoxy in va uum

Within the frequen y range of al ulations, su h a omposite material an be onsidered as an orthotropi

homogeneous medium for whi h the equivalent density is

ρ

“1560 kg¨m

´3

. The rystallographi dire tions

71

,

72

and

73

orrespond to the

x

,

y

and

z

-axis, respe tively. The dire tion of bers oin ides with the

x

-axis.

This ompositematerialhas been hosen be auseit has already been used insome experiments [16,Table 2.3℄.

Plate thi kness is

3.6 mm

. Thehomogenizedstiness onstantsare:

c

11 “86.65

,

c

22 “14.00

,

c

33 “14.00

,

c

44 “3.00

,

c

55 “4.06

,

c

66 “4.70

,

c

12 “7.50

,

c

13 “7.50

,

c

23 “7.00 rGP as

. The sour e is lo ated on the rst interfa e

z

“ 0

. In

agreementwith the experiments, the normalfor e atthis surfa e, dened inEq. (1), is modeledby aGaussian

distributed amplitude inthe

xy

-plane and oupledto atone burst signal, su hthat:

σ

0 px, tq “ exp

ˆ

´

x

2 ` y

2

2 σ

2 ˙

sin

_pω

0 t

q exp

ˆ

´

ω

2

0 t

2

2 n

2 c

˙

`

0 0 1

˘

T

,

(21)

with

ω

0 “ 2 π f

0

. The other new variables are: the Gaussian aperture

σ

“ 2.4 mm

, the entral frequen y

f

0 “

150 kHz

andthenumberof y les

n

c

“ 6.5

. Firstofall,letusexaminedispla ementontheplatesurfa e,i.e.,at

z

_{“ 0}

,foraxedtime

t

“ 120 µs

. Thisisarelativelylongtimeforguidedwavestopropagateoverlongdistan es. The mat hedsurfa e of al ulationis su hthat:

x

P p´1.12 .. 1.12 m, 1030s

and

y

P p´0.39 .. 0.39 m, 360s

, where

pv

min

.. v

max

, n

s

denotes a half- losed dis retized interval of

n

linearly spa ed values from

v

min

to

v

max

. This

notation will be used for all sampled variables introdu ed later on. Thus, the waveforms are observed for

t

_{P r´22 .. 228 µs, 150q}

. For this 3D problem and for this large plane of analysis, the omputation time is

25

1

,

whi hisarelativelyshorttimethat ouldbe onsiderablyredu edbyusingafaster omputer. Asanillustration,

Fig.3-arepresentstheverti aldispla ement

u

z

foranon-dissipativeplate. Inthefrequen y rangeoftheemitter

and for a omposite sample,onlyfour propagatingmodes an exist. Thethree fundamentalmodes

A

0

(exural

wave),

SH

0

(horizontal shear wave) and

S

0

( ompressional wave), and the rst anti-symmetri mode, usually

alled

A

1

. The three fastest fronts orrespond tothe three fundamentalmodes

A

0

,

SH

0

and

S

0

. It an beseen

in [7℄ that there is good agreement between the ray theory and the eld al ulation by using a time-domain

modalapproa h. Inthis paper,the waveforms have beenobtained withoutmodalde ompositionbut, of ourse,

they orrespond exa tly tothe waveforms obtained in[7℄, sin e both are exa t solutionsof the same problem.

The theory is now ompared with experiments reported in [16℄. Forthis, two waveforms are al ulated attwo

dierentpositionsonthesurfa e,

p15, 0, 0q

and

p21.05, 17.68, 0q

,whi hareidentiedby thetwodotsinFig.3-a.

The asso iated waveforms are shown in Figs. 3-b and 3- . Clearly, the agreement between experimental and

theorywavefrontarrivaltimesis orre t, whiletheamplitudesdierdrasti ally. Thisisduetowavedamping,as

iswell-knowninsu h omposites[10℄. Totakethisee t intoa ount,inagreementwithexperimentsperformed

in the frequen y range asso iated with the sour e des ribed by Eq. (21), the vis oelasti model, dened by

Eq. (20), is used. The asso iated vis osity onstants of the homogenized solid are:

η

11 “ 2.00

,

η

22 “ 0.30

,

η

33 “ 0.30

,

η

44 “ 0.085

,

η

55 “ 0.085

,

η

66 “ 0.055

,

η

12 “ 0.13

,

η

13 “ 0.13

,

η

23 “ 0.106 rkP a¨rad

´1

_s

. The values

of these onstantsdier fromthose given in [16℄. This is be ause the values obtained fromexperiments inthis

paper, asmentioned, in ludenot onlyvis osity absorption, but also wave attenuation due to beam dira tion.

(12)

0

20

40

60

80

100 x

[

cm

]

0

5

10

15

20

25

30

35 y

[

cm

]

t

= 120 µs

SH

0,1

SH

0,2

A

1 A

0 S

0 a-1

No dissipation

0

20

40

60

80

100 x

[

cm

]

t

= 120 µs

SH

0,1

SH

0,2

A

1 A

0 S

0 b-1

Hysteretic model

0

50

100

150

200 t

[µs]

−16

−12

−8

−4

0

4

8

12

16 a

m

p

lit

u

d

e

a

rb

.

u

n

it

s

₀

_°,

_{15 cm}

A

0 S

0 a-2

0

50

100

150

200 t

[µs]

0 °,

15 cm

_A

0 S

0 b-2

50

100

150

200 t

[µs]

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5 a

m

p

lit

u

d

e

a

rb

.

u

n

it

s

40 °,

27.5 cm

S

0 SH

0,1

SH

0,2

A

0 a-3

50

100

150

200 t

[µs]

40 °,

27.5 cm

S

0 SH

0,1

SH

0,2

A

0 b-3

Figure 3: 3D response (verti aldispla ement)of amonolayerplate of arbon-epoxyto asour e lo atedat thetop

interfa e(

z

“ 0

). (a)Non-dissipativemedium. (b)Dissipation onsideredthroughthehystereti model. Figures(a)-1 and(b)-1 are views from thetopof theplate after alongtime, with non-lineargrays ale in orderto showthelow

amplitude

SH

0

and

A

1

modes. Ingures(a)-2,3and (b)-2,3thenumeri al omputations(redsignals)are ompared totheexperimentaldata(bla ksignals)[16℄.

sin e beam dira tion is, of ourse, taken into a ount by the model. The omparison between theoreti al

waveforms obtained for a vis oelasti plate and experimental signals is presented in Fig. 3. For the two angles

of observation, note the very goodagreementbetween theory and experiment.

4.2 Aluminum plate immersed in water

To illustrate the e ien y of the method, it is of interest to al ulate the response near the sour e. To this

end, let us onsider an aluminum plate immersed in water, whi h is insonied by a transdu er near a ZGV

(Zero Group Velo ity) ondition. Re ently,new methodsof NDT have been developed taking advantage of the

sensitivitytotheinterfa eparametersofwavesgeneratedinZGV onditions (e.g.,[21℄). Inour ase,the sample

thi kness is

1 mm

and the density is

ρ

1 “2780 kg¨m

´3

. Using the isotropy relations, all the stiness onstants

are dedu ed from the two independent stiness onstants:

c

11 “112.0

and

c

66 “27.0 rGP as

. The water density

is

ρ

0 “ 1000 kg¨m

´3

and the sound velo ity inwater is

1500 m

¨s

´1

. Toobserve the ZGVphenomenon learly,it

is importantto minimizethe wave dira tionee ts and togenerate aspe i Lambwave pre isely ata single

frequen y. To do this, the simulationof the transdu er, whi h is lo ated in the upper semi-innite layer, i.e.,

β

_{“ 0}

in Eq. (1), is given by the followingfun tion:

f

0 px, y, z, tq “ Π

„

2 _{pz ´ z}

f

q

l sin θ



exp

#

´

rx ´ x

0 pzqs

2

2 σ

2 x

´

y

2

2 σ

2 y

+

sin

_pω

0 t

q exp

ˆ

´

ω

2

0 t

2

2 n

2 c

˙

¨

˝

sin

_pθq

0 cos

_pθq

˛

‚,

(22)

where fun tion

Π

is the re tangular fun tion su h that

Π

pξq “ 1

if

|ξ| ď 1{2

, and

Π

pξq “ 0

otherwise. The

quantity

x

0 pz, θq

, given by

x

0 pz, θq “ pz ´ z

f

q cotpθq ´ z

f

tan

pθq

, expresses the transdu er rotation around its enter at position

z

f

, with an additional lateral shift to make the transdu er entral axis ross the rst

(13)

θ

f

0

2.6

°

3.0 MHz

6.0

°

2.87 MHz

12.7

°

3.0 MHz

Table 2: The three

pθ, f

0 q

pairsused in formula (22)to sele tthe ba kwardpropagative,zeroenergy velo ity and propagativeregimes.

interfa e at the referen e origin dened in Fig. 4. The variable

l

stands for the length of the transdu er. The

other variables are the same as those dened in Eq. (21). The parameter values are su h that:

σ

x

“ 0.5 mm

,

σ

y

“ 10.0 mm

,

l

“ 20.0 mm

,

z

f

“ ´6.0 mm

and

n

c

“ 30

. Be ause the sour e

f

0 px, y, z, tq

is not proportionalto

a Gaussian fun tion rotated by the angle

θ

, it does not model exa tly a xed transdu er that is rotated in a

uid. However, resultsare very losetoauniquerotatedtransdu er,andthe analyti al al ulationsaresimpler.

These values orrespond to a 3D transdu er that generates a quasi-plane wave in

θ

-dire tion. For the three

ases under onsideration,the valuesof

θ

and

f

0

are given inTable 2. Asthe angle ofin iden e in reases,these

ases orrespond to a negative group velo ity, the ZGV ondition asso iated to

S

1

mode, and apositive group

velo ity respe tively. The onvolutionexpressed by Eq. (A.12) is al ulated analyti ally.

−4

−2

0

2

4 x

[cm]

−4

−2

0

2

4 x

[cm]

−4

−2

0

2

4 x

[cm]

12

14

16

18

20 ti

m

e

[µ

s]

12.7°

6.0°

−4

−2

0

2

4 y

[c

m

]

2.6°

Figure4: Verti aldispla ement

u

z

atthesurfa eofthealuminumplateforthethreein iden eangles(seeTable2): (a) for

t

“ 19 µs

with respe t to

x

and

y

; (b) for

y

“ 0

with respe t to

x

and

t

; the bold signal orresponds to

t

“ 19 µs.

Fromlefttoright: theba kwardpropagative,zeroenergyvelo ityandpropagativeregimes.

To perform al ulations by satisfying the Nyquist riterion, the plate surfa e has been sampled with

x

P

p´60 .. 60 mm, 270s

and

y

P p´60 .. 60 mm, 16s

. Azero paddingpro edureis thenapplied tothe al ulated eld spe trum to in rease the denition of the a ousti elds. The time sampling is su h that:

t

P r0 .. 20 µs, 200q

.

For this quasi 2D problem, the al ulation time is

5

2

. Results are presented in Fig. 4-a, where the verti al

displa ements at the plate surfa e, i.e. at

z

“ 0

, are plotted in gray s ale, for a xed time

t

“ 19 µs

, for the

three spe ied ases. In Fig. 4-b, the same displa ement, obtained for

y

“ 0

and

z

“ 0

, is shown for some

observation times versus

x

-positions. These graphs start at

12 µs

, sin e this time is approximately when the

in ident wave vanishes and the free regime appears. The bold urves orrespond to the xed time of

19 µs

. As

expe ted, itis learlyobserved that forthe ZGVanglethe guidedwavedoesnot propagatealongthe plate,but

the transmitted beam is on entrated in the plate under the in ident beam, whi h insonies the plate around

(14)

dire tion of the group velo ity. Of ourse for any angle of in iden e, as time in reases, the wave amplitude

de reases moreor less due to radiationleakage into the water.

4.3 Plate made of 8 sta ked arbon-epoxy layers immersed in water

As anal example, letusinspe t the responseof a multilayered plateimmersed in water. The plate onsists of

eight sta ked arbon-epoxy layers. Ea h layer ismade of the arbon-epoxy omposite des ribed in se tion4.1,

for whi h vis oelasti ityis taken intoa ount by Eq. (20). Itsthi kness is

3.6 mm

. The rotationsequen e is as follows: [0°/135°/90°/45°/0°/135°/90°/45°℄.

Figure5: Relativeamplitudeoftheradialsour edenedbyEq.(23),distributed inlengthanddepth.

To analyze the method possibilities in a new ontext, the sour e now represents a line radial impulse lo ated

inside the plate. It is distributed horizontally and with depth, as shown in Fig. 5, and is invariant in the

y

-dire tion. It is entered at depth

z

6 “ 21.6 mm

and has ahalf width of about the thi kness of one layer, and therefore extends overfour layers.

Although partial waves an exhibit three non-zero omponents, the problem is then redu ed to a 2D spa e

problem. The sour e expression, inagreement withthe denition given inEq. (1), is su h that:

f

β

px, y, z, tq “ exp

«

´

x

2 ` pz ´ z

6 q

2

2 σ

2 ﬀ

exp

ˆ

´

t

2

2 σ

2 t

˙

_`

x 0 z

_{´ z}

6 ˘

T

,

(23)

with

σ

“ 1.5 mm

and

σ

t

“ 0.25 µs

. Thevalueoftheparameter

σ

hasbeen hosentoexpandtheinitialfor eonly onthefourlayersidentiedby

β

“ 5, 6, 7, 8

. Thelateralpositionissampledwith

x

P p´250 .. 250 mm, 370s

. The onvolution(A.12)is al ulatedbyusingatrapezoidalrulewiththefollowingsampling

z

P r15.6 .. 27.6 mm, 101s

. From a numeri alpoint of view, be ausethe ex itation ontains nullfrequen y in this ase, itis ru ialtodeal

with vis oelasti ity orre tly. As a matter of fa t, non-respe t of ausality an provide errors in al ulating

waveform. The two proposed models do not exhibit the same behavior in this regard,sin e the model dened

by Eq. (20) is not ausal, while using Eq. (19) ensures ausality. This non- ausality omes from the model

behavior atlowfrequen ies. Letus sampletime by

t

P r0 .. 40 µs, 240q

. This sampling isoptimal torespe t the

Nyquist-Shannon theorem, using the same rule as before. The waveforms must therefore be well al ulated.

However, this is not the ase for the non- ausal model, as shown in Fig. 6-b-1, where waveforms are a quired

at position

p0, 0.6q

in the

xz

-plane. Comparing the non-dissipative plate (solid line) to the dissipative plate

(dashed line), reveals substantial dieren es. Indeed, adverse ee ts appear for long times, where the signal

amplitude in reases exponentially. Let us give the explanation. Non respe t of ausality generates nonzero

signal amplitude for low negative times. Consequently, the periodization inherent in spe tral dis retization

reates a opy of this signal at times lose to the end of the observation window. After pro essing with the

Lapla einversionalgorithm,this ontributionis onverted toanexponentialin rease thatstartsafter

30 µs

. To

avoidthis problem,it isthen ne essary to in rease the observation windowas wellas the numberof points. As

a onsequen e, time isnow sampled with

t

P r0 .. 60 µs, 360q

. The orre ted waveform, for whi h the last

20 µs

are ignored,is plottedin Fig. 6-b-1 (pointed line). This urve isnot identiable be ause the observation point

(15)

paq

pbq

0.0

2.88 t

= 2.0 [µs]

0.0

2.88 t

= 6.0 [µs]

0.0

2.88 t

= 10.0 [µs]

−2

0

2

4

6

8

10

12 x

[cm]

0.0

2.88 t

= 35.0 [µs]

−2

0

2

4

6

8

10

12 x

[cm]

−15

−10

−5

0

5

10 −2

−1

0

1 d

is

p

la

c

e

m

e

n

t,

a

rb

.

u

n

it

s

0

5

10

15

20

25

30

35

40 t

[µs]

−1.0

−0.5

0.0

0.5 z

[c

m

]

Figure6: (a)Verti aldispla ement

u

z

for

t

“ 2, 6, 10

and

35 µs

,forahystereti dissipationlawand omputedwith the "in reased" time window. (b)-2,3 Signals for the dissipativeplate (solidline) and for thenon-dissipative plate

(dashedline);thesesignalsare omputedatthepositionsindi atedbylight(yellow)dotsinthe orrespondinggures

(a). (b)-1A thirdsignalplottedwithahalf-dashedline orrespondstothedissipativeplateandthe"non-in reased"

time window, and reveals the behavior of the numeri al artifa t growing exponentially over time. (b)-4 Verti al

displa ementuzat

t

“ 35 µs

, omputedusingthe"non-in reased"timewindow: anumeri alartifa t anbeobserved duetonon-respe tof ausality;itismaximalatthelo ationofthesour etermandattheendofthetimewindow.

is very lose to the sour e, and there is therefore no dieren e between dissipative and non-dissipative plate

responses.

Let us now examine all the results for this dissipative ase. They are presented in Figs. 6-a-1 to 6-a-4, where

images of verti al displa ement

u

z

are shown at four xed times (

t

“2, 6, 10, 35 µs

). As time in reases, leaky

Lamb waves are seen to form. These guided waves reate line wavefronts that radiate energy in water. In

addition, for the longest time, the displa ement eld

u

z

, al ulated with the non-mat hed sampling, is shown

in g. 6-b-4. The artifa t mentioned above appears at the lo ation of the sour e. In order to appre iate

the vis oelasti ee ts better, three waveforms a quired at three xed positions in the

xy

-plane are plotted

in Figs. 6-b-1 to 6-b-3. The three positions are lo ated at:

p0, 0.6q

,

p2.72, 1.25q

and

p6.66, 2.75q

and they are identied by light(yellow) dots in Figs. 6-a-1 to6-a-4. Thewaveforms obtained fornon-dissipativeplate (solid

line) are ompared with those al ulated for dissipative plate (dottedline). Clearly,the vis oelasti ee ts are

not reallyimportantin omparisonwith resultsdes ribed inse tion 4.1. Thisis be ausethe spe tral ontent is

mainly on entrated at lowfrequen ies.

Beyondtheseexpe tedphysi alobservations,thisstudy illustratesthee ien yofthe methodwhi h omputes,

ina reasonabletime, the exa t a ousti response toanextended sour eina multilayeredimmersed plate. This

plate mayormaynot bevis oelasti . Indeed,with this sampling,to al ulatethe a ousti eld inthe

xy

-plane

and atany time,

25

2

(

38

2

forvis oelasti plate) are ne essary. To on lude this se tion,itisof greatinterestto

(16)

In future work, this possibility willbe used to study the dira tion of guidedwaves by defe ts.

5 Con lusion and prospe ts

Wehave adaptedanexistingmethod,toolittleusedinultrasoni s, for omputingultrasoni sour eintera tions

withmultilayeredplates,basedontimeanalysisusingtheLapla etransform. Thismethodprovidesausefultool

whi h omplementsthemu husedte hniquebasedongeneralizedLambwavede omposition. Byavoidingmode

analysis whi h an be problemati in some ases exa t numeri al al ulations an be made in a relatively

short time for immersed plates and vis oelasti layers. Even for 3D ases, numeri al osts are relativelylow.

These al ulationswillbeusedtodevelopthetopologi alimagingofdefe tsdeeplyhiddenatsub-sour epositions

in multilayered stru tures. The topologi al imaging method is based on intera tions between two al ulated

elds, derived from the so- alled dire t and adjoint problems (e.g., [3,5, 22℄, and [25, 27℄ in geophysi s). The

tool that we develop will be of prime interest to image defe ts in 3D omposite materials in quasi-real-time,

both alongthe plate and normalto the surfa e.

There are no spe i restri tions on erning the size and the spe trum width of sour es, ex ept, of ourse,

Shannon's sampling onditions. As a result, the Green tensor an be al ulated without additionaleorts and

anbeusedtobuildaBoundaryElementMethod. Thispossibilityhasbeensu essfully onsideredformodeling

the intera tionofplanardefe tsinlayeredwaveguideswithatransientin omingwave[15℄. Anextensionisnow

underway tobuild aDiri hlet-to-Neumannoperatorthat willenable this tool tobe oupledwith a ommer ial

Finite Elements software.

A knowledgments

This work is jointly funded by Region Aquitaine and CEA-List, Fran e.

Referen es

[1℄ Audoin B., Guilbaud S., A ousti waves generated by a line sour e in a vis oelasti anisotropi medium,

Appliedphysi s letters, 72(7)(1998) 774-776.DOI 10.1063/1.120889

[2℄ AuldB.A., A ousti Fields and Wavesin Solids,se ond ed., Krieger Pub. Co., 1990. ISBN0894644904

[3℄ BonnetM.,GuzinaB.,Soundingofnitesolidbodiesbywayoftopologi alderivative,Int.J.Numer.Methods

Eng. 61 (2004)23442373. DOI 10.1002/nme.1153

[4℄ Bonnet-Ben Dhia A.-S., Goursaud B., Hazard C., Prieto A., Finite element omputation of leaky modes in

stratied waveguides,in: Leger A.,Des hamps M.,(Eds.), Ultrasoni wavepropagation in non-homogeneous

media, Springer pro eedings in physi s, vol. 128. Springer, Berlin,2009, pp. 73-86. ISBN 3540891048

[5℄ Dominguez N., Gibiat V., Non-destru tive imaging using the time domain topologi al energy method,

Ultrasoni s50 (2010) 367372.DOI 10.1016/j.ultras.2009.08.014

[6℄ Du asse E., Des hamps M., A nonstandard wave de omposition to ensure the onvergen e of Debye series

for modeling wave propagation in an immersed anisotropi elasti plate, Wave Motion 49 (2012) 745-764.

(17)

[7℄ Du asseE.,Des hampsM.,Time-domain omputationoftheresponseof omposite layeredanisotropi plates

to a lo alized sour e, Wave Motion 51 (2014) 1364-1381. DOI 10.1016/j.wavemoti.2014.08.003

[8℄ Gavri¢ L., Computation of propagative waves in free rail using a nite element te hnique, J. Sound Vib.

185(3)(1995) 531-543.DOI doi:10.1006/jsvi.1995.0398

[9℄ Hayashi T., Inoue D., Cal ulation of leaky Lamb waves with a semi-analyti al nite element method,

Ultrasoni s54(6)(2014) 1460-1469. DOI 10.1016/j.ultras.2014.04.021

[10℄ Hosten B., Des hamps M., Tittmann B. R., Inhomogeneous wave generation and propagation in lossy

anisotropi solids. Appli ation to the hara terization of vis oelasti omposite materials, J. A oust. So .

Am.82(5) (1987)1763-1770. DOI 10.1121/1.395170

[11℄ Kausel E. and Roësset J., Frequen y domain analysis of undamped systems, J. Eng. Me h., 118(4) (1992)

721-734.DOI 10.1061/(ASCE)0733-9399(1992)118:4(721)

[12℄ Kausel E., Thin-layer method: formulation in the time domain, Int. J. Numer. Meth. Eng. 37(6) (1994)

927-941.DOI 10.1002/nme.1620370604

[13℄ Liu G.R.,TaniJ., Ohyoshi T., WatanabeK.Chara teristi wave surfa es in anisotropi laminated plates,

J.Vib. A oust.,113(3) (1991)279-285. DOI 10.1115/1.2930182

[14℄ LoweM.J.S.,Matrixte hniquesformodelingultrasoni wavesinmultilayeredmedia,IEEETrans.Ultrason.

Ferroele tr. Freq.Control42(4) (1995) 525-542.DOI 10.1109/58.393096

[15℄ MoraP.,Du asseE.,Des hampsM.,Intera tion of aguidedwave witha ra kin anembedded multilayered

anisotropi plate: global matrix with Lapla e transform formalism, Pro eedings of ICU 2015, Physi s

Pro edia 70(2015) 326-329.DOI 10.1016/j.phpro.2015.08.217

[16℄ Neau G., Lamb waves in anisotropi vis oelasti plates: study of the wave fronts and attenuation (Ph.D.

thesis),Bordeaux I University, 2003.

[17℄ Park, J. Wave motion in nite and innite media using the Thin-Layer Method (Ph.D. thesis),

Massa husetts Instituteof Te hnology, 2002.

[18℄ ParkJ.,KauselE.,Response of layered half-spa e obtaineddire tlyin the timedomain, part 1: SHsour es,

Bull.Seismol. So .Am. 96(5)(2006) 1795-1809. DOI 10.1785/0120050247

[19℄ Phinney R. A., Theoreti al al ulation of the spe trum of rst arrivals in layered elasti mediums, J.

Geophys. Res. 70(20) (1965) 5107-5123. DOI 10.1029/JZ070i020p05107

[20℄ Pon elet 0., Des hamps M. , Lamb waves generated by omplex harmoni inhomogeneous plane waves, J.

A oust. So . Am.102(1) (1997)292-300.DOI 10.1121/1.419752

[21℄ Prada C., Clorenne D., RoyerD., Lo alvibration of an elasti plate and zero-group velo ity Lamb modes,

J.A oust. So .Am. 124(1)(2008) 203-212.DOI 10.1121/1.2918543

[22℄ RodriguezS.,Des hampsM.,CastaingsM.,Du asseE.,Guidedwavetopologi alimagingofisotropi plates,

Ultrasoni s54 (2014) 18801890.DOI 10.1016/j.ultras.2013.10.001

[23℄ S hmidt H., Tango C., E ient global matrix approa h to the omputation of syntheti seismograms,

Geophys. J. Roy.Astr. So .84(2) (1986)331-359. DOI 10.1111/j.1365-246X.1986.tb04359.x

[24℄ Stroh A.N., Steady state problems in anisotropi elasti ity, J. Math. Phys. 41 (1962) 77103.

DOI 10.1002/sapm196241177

[25℄ Tarantola A., Inversion of seismi ree tion data in the a ousti approximation, Geophysi s 49(8) (1984)

(18)

[26℄ Veli hko A., Wil ox P. D., Modeling the ex itation of guided waves in generally anisotropi multilayered

media, J. A oust. So . Am.121(1) (2007) 60-69.DOI 10.1121/1.2390674

[27℄ Virieux J.,Operto S., An overview of full-waveform inversion in exploration geophysi s, Geophysi s 74(6)

(2009)WCC127WCC152.DOI 10.1190/1.3238367

[28℄ WeaverR.L.,Sa hse W. andNiu L.,Transient ultrasoni waves in a vis oelasti plate: Theory, J.A oust.

So .Am. 85(6)(1989) 2255-2261.DOI 10.1121/1.397770

Appendix A On the use of the Green tensor of an innite anisotropi

medium, in the

pk, z, sq

-domain

A.1 General onsiderations

A sour e emitting in the medium labeled

β

and lo ated between

z

“ z

min

and

z

“ z

max

is represented by the

followingterm:

f

β

px, z, tq “

ż

z

max

z

min

δ

_{pz ´ ζq f}

β

px, ζ, tq

d

ζ .

(A.1)

In the

pk, z, sq

-domain,the response of the medium, onsidered as aninnite spa e, isexpressed asfollows:

˜

U

_{pzq “}

ż

z

max

z

min

˜

G

_{pz ´ ζq ˜}

F

β

pζq

d

ζ .

(A.2) where

˜

G

denotes the Green tensor.

In the

pk, k

z

, s

q

-domain,the Green tensor is simplyexpressed asfollows:

ˆ

G

_{“ ˆ}

A

´1

, where ˆ

A

_“

“

ρ s

2

I

`pK˛Kq

‰

“

ρ s

2

I

` pk˛kq ` k

z

“

pn˛kq`pk˛nq

‰

`k

2 z

pn˛nq

(

,

(A.3)

and is ne essarily holomorphi onthe omplexhalf-plane

Re

psq ą 0

.

A.2 Green tensor in the

pk, z, sq

-domain

The followingFourier transform yieldsthe Green tensor inthe

pk, z, sq

-domain:

˜

G

_{pzq “}

1

2 π

ż

`8

´8

exp

_p´

i

k

z

q ˆ

A

pk, k

z

, s

q

´1

d

k

z

,

(A.4)

The determinant of the matrix

ˆ

A

_{pk, k}

z

, s

q

dened by Eq. (A.3) is a polynomial fun tion of

s

2

(third degree)

and

k

z

(sixth degree) with real oe ients and is usually named the Christoel polynomial. Consequently,

the Green tensor

ˆ

G

_{pk, ‚, sq}

has exa tlysixpoles

k

z,i

pk, sq

in the omplexplane,whi hare ontinuous fun tions of the Lapla e variable

s

.

If

Re

psq ą 0

,

k

z,i

pk, sq

anneverbearealnumberbe ausethematrix

ˆ

A

_{pk, k}

z

, s

q´ρ s

2

Iisreal-valuedsymmetri

positive-denite if

k

z

isalso areal number.

If

s

is apositivereal number, the matrix

ˆ

A

_{pk, k}

z

, s

q

is real-valued symmetri positive-denite, and the verti al

wavenumbers

k

z,i

pk, sq

ne essarily appear in omplex onjugatepairs. Thus, there are three wavenumbers with

(19)

This latter property an be extended to

s

values in the half-plane

Re

psq ą 0

be ause the imaginary part of

k

z,i

pk, sq

isa ontinuous fun tion of

s

whi h annotbezero on the omplex half-plane

Re

psq ą 0

. Furthermore,

ˆ

G

_{pk, k}

z

, s

q

∼

k

z

Ñ˘8

k

_z

´2

_{pn ˛ nq}

´1

.

A ordingly,by applying theresidue theoremonanhalf-diskof radiustendingtoinnity, the Greentensor an

be expressed asfollows:

˜

G

_{pzq “}

ˇ

3 ÿ

i“1

exp

_p´

i

k

z,i

z

q ˜

B

i

if

z

ă 0 pupgoing wavesq

6 ÿ

i“4

exp

_p´

i

k

z,i

z

q ˜

B

i

if

z

ą 0 pdowngoing wavesq

(A.5)

Furthermore, if the Green tensor asso iatedto the verti alstress isintrodu ed asfollows:

˜

T

_{pzq “ ´}

i

pn ˛ kq ˜

G

pzq ` pn ˛ nq B

z

˜

G

_{pzq ;}

(A.6)

it an be demonstrated that

G

˜

and

T

˜

satisfy the followingordinary dierential system:

B

z

„

_˜

G

_pzq

˜

T

_pzq



“

„

i

pn˛nq

´1

_pn˛kq

_pn˛nq

´1

ρ s

2

I

` pk˛kq ´ pk˛nq pn˛nq

´1

_pn˛kq

i

pk˛nq pn˛nq

´1



looooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooon

S

„

_˜

G

_pzq

˜

T

_pzq



´ δpzq

„

O I



(A.7)

Consequently, the six omplex verti al wavenumbers

k

z,i

are the eigenvalues of the matrix i

S

, proportional to the so- alled Stroh Matrix [24℄. They are asso iatedto eigenve tors

ξ

i

su h that:

„

_˜

G

_pzq

˜

T

_pzq



“

ˇ

3 ÿ

i“1

exp

_p´

i

k

z,i

z

q ξ

i

q

T

i

if

z

ă 0 pupgoing wavesq

6 ÿ

i“4

exp

_p´

i

k

z,i

z

q ξ

i

q

T

i

if

z

ą 0 pdowngoing wavesq

(A.8)

where the 3D ve tors

q

i

satisfy the following invertible linear system of 18 equationswith 18 unknowns:

3 ÿ

i“1

ξ

_i

q

T

i

´

6 ÿ

i“4

ξ

_i

q

T

i

“

„

O I



.

(A.9)

A.3 Response to an extended sour e

The response toanextended sour elo atedbetween

z

“ z

min

and

z

“ z

max

isobtained by ombiningEqs.(A.2)

and (A.8):

˜

H

_{pzq “}

„

_˜

U

_pzq

˜

Σ

z

pzq



“

ż

z

max

_pz,z

min

q

3 ÿ

i“1

exp

_r´

i

k

z,i

pz ´ ζqs ξ

i

q

T

i

F

˜

β

pζq

d

ζ

`

ż

min

_pz,z

max

q

z

min

6 ÿ

i“4

exp

_r´

i

k

z,i

pz ´ ζqs ξ

i

q

T

i

F

˜

β

pζq

d

ζ .

(A.10)

An observation pointbetween

z

min

and

z

max

re eivesboth upgoing wavesfromthe part of the sour e belowthe

pointand downgoingwaves from the part of the sour e above the point: