AN EFFICIENT MODEL TO PREDICT GUIDED WAVE RADIATION BY FINITE-SIZED SOURCES IN MULTILAYERED ANISOTROPIC PLATES WITH ACCOUNT OF CAUSTICS

(1)

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AN EFFICIENT MODEL TO

PREDICT GUIDED WAVE

RADIATION BY FINITE-SIZED

SOURCES IN MULTILAYERED

ANISOTROPIC PLATES WITH

ACCOUNT OF CAUSTICS

AFPAC 2015 – FREJUS | Mathilde Stévenin

Alain Lhémery

Sébastien Grondel

(2)

General field expression

Finite-sized sources

(3)

• Many advantages

•

Long distance propagation

•

Fast inspections of large structures

(fixed transmitter and receiver)

• But difficult interpretation of inspection results

•

Multi-modal

(several modes with different speeds

at one given frequency)

•

Dispersive

(speeds depend on frequency)

• Target applications:

• Non-Destructive Evaluation

• Structural Health Monitoring

• Acoustical Emission

• …

e = thickness 20 mm

f =

frequency 0,2 MHz

e xf [MHz.mm]

BACKGROUND:

Guided waves non destructive testing of plate-like

structures

AFPAC 2015 – FREJUS | Mathilde Stévenin

In all cases transducers of finite size

are used

Phase velocity dispertion curve

Pha se vel oci ty (km /s ) 3/24

(4)

• _{in multilayered anisotropic plates: composite applications}

• efficient: computation time compatible with industrial use of simulation

• taking into account the caustics

• modal solution: easier result interpretation

OBJECTIVES:

An efficient model to predict guided wave radiation by

finite-sized sources

• Modal solution by the Semi-Analytical Finite Element (SAFE) method [1] for

each direction

• Finite sized sources:

• isotropic plates

• Fraunhofer-like approximation for radiation by finite-sized sources [2]

• multilayered anisotropic plates

• numerical methods but high computational cost

THE BASES:

[1] Taupin, Lhémery and Inquité,

J. Phys. : Conf. Ser.,

269 , 012002 (2011).

[2] Raghavan and Cesnik,

Smart Mater. Struct.,

14 , 1448 (2005).

(5)

General field expression

Finite-sized sources

(6)

• Convolution of the source and a Green’s function

•

With :

u

(3)

_{, displacement}

g

(3)

, Green’s function

q

(3)

_{, source}

•

Modal Green’s function :

n is the number of phase contributions for a given observation direction and a given mode m

q













(3)

, , ,

',

', '

'

S

x y z q





x



x y



y z

x y dx dy

u

g

q









(3) (3) , ( , )

, ,

_{m n}

, ,

m n m

x y z





 

g

(7)

SH0 mode

[0°/90°]

_S

T700GC/M21 cross-ply composite fiber-reinforced polymer

f = 300kHz

for this energy (observation) direction three phase contributions

(8)

FAR FROM CAUSTICS

phase approximation thanks to three parabolas (second order approximation)

and then sum of the three contributions

(9)

















 

_{ }

 





2 , , 2 2 2 , , , , , sgn 1/ 2 ₄ , , , (3) 2 2 ₂ , 2 , , 2

, ,

,

2

m n m n m n m n m n m n i i _{m n} _{m n} i x y m n _{k k} m n m n

k

g

x y z

x

y

e

e res G

   _    









_{ }     _{ } _ _ _   





 



• Expression of the modal contribution far from caustics

• Calculated thanks to stationary phase method [1]

With ,

phase term,

G

spatial Fourier transform of

g

,

wavenumber calculates thanks to the SAFE method



,



  

, ,



, m n

,

, m n

cos

m n





k

m n











[

1] Velichko and Wilcox,

J. Acoust. Soc. Am.,

121 , 60 (2007).

 

tan

y

x





, m n

k

(10)

NEAR CAUSTICS

phase approximations:

-

parabolic approximation

(11)

• Expression of the modal contribution near caustics

• Calculated thanks to stationary phase method [1]

With

, ,

, and

Ai

Airy function









 

_{ }

 

_{ }

 

_

 

_

 

_{ }

 

_

_

2 2 ,1 ,1 ,2 ,2 1/3 2 (3) 2 2 ,1 2 ,1 ,2 , ,1 ₂ ,2 ,2 ₂ , ,1 , ,2 2 2

, ,

2

2 ,

,

m m m m i i x y L m m m m n m m m k k k k m n m m n m

e

g

x y z

x

y

e

Ai

S

res G

k

res G

k

   













    











_









_



_

 







_







[1] Karmazin, Kirillova, Seeman and Syromyatnikov,

Ultrasonics ,

53 , 283 (2013).

 

,1

_

,1

_

,2

_

,2

_

1 ,

,

2

m m m m

L





_

_

















_

_

 









2

3 ₃

,

, 2

,1

4 S

m







_



_







 







_



_











2 2



23

 

x

y

S



 





(12)

General field expression

Finite-sized sources

(13)

ISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION

(x_s,y_s) a

(x_s,ys) LX

LY

Simplification of the displacement field expression, thanks to the isotropic

properties of the material:





 







 







  2 2 (3) (3) ₄ (3) 1/ 2 2 2 ' '

, , ,

,

2 '

'

m m i m s s k k m ik x x y y S

k

x y z q

e res G

x y

x

y

e

dx dy



    















u

q



 





2 2



1/2



 

2



2



1/2

'

_s _s

x

y

x

y

 













 





 









 













 







2 2 2 2 2 2 2 2 1 ' ' ' ' s s m s s s s s s m x x y y ik x x y y x x y y x x y y x x y y ik x x y y

e

 _ _       _     _  _ _{ } _ _{ }     _ _ 13/24

(14)







 







     

_{ }

 

_{ }

_

_

2 2 (3) (3) 1/ 2 2 2 ₍₃₎ 4 ,

, , ,

,

2

s s m m m m i i x x y y k m m s s m fraun s s m k k

x y z q

k

x

y

e

e res G

F

x y

  





    











u

q

ISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION

(xs,ys) a (xs,ys) LX LY

 

 





 



 





 



,

₂ ₂ ₂ ₂

2

m m s X m m s Y m rect X Y s s s s

k

x

L

k

y

L

F

x y

L L sinc

sinc

x

y

x

y





_





_















_

_

_





_

_

_











 

2 1



_{ }

 



,

2 ,

m m m disc

J k

a

F

x y

a

k

a









(15)

ANISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION

• Plate characteristic

• [0°/90°]

_S

T700GC/M21 cross-ply composite

fiber-reinforced polymer

• Plate thickness: 1mm

C

₁₁

(GPa)

C

₂₂

=C

₃₃

(GPa)

C

₁₂

=C

₁₃

(GPa)

C

₂₃

(GPa)

C

₄₄

(GPa)

C

₅₅

=C

₆₆

(GPa)

Mass

density

(kg/m

3

₎

Ply

thickness

(mm)

123.4

11.5

5.6

6.4

2.6

4.5 1.6x10

3

_0.25

f = 300kHz



x

y

15/24

(16)

Observation points

Finite-sized source

Computed results:



100 ,



z

u

R



mm



(17)

Convolution

Fraunhofer-like approximation

ANISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION

Normal displacement comparison

Disc shaped transducer of radius a=5mm

Observation distance100mm

Even for the less anisotropic

mode: approximation fails

A0 mode

(18)

INTEGRATION ALONG ENERGY (OBSERVATION) DIRECTIONS

θ

Integration along segments (change

of variables in surface integral)





 





_

_

2 max min 1 ' , (3) (3)

1

(3) (3)

0,

0, ,

,

',

'

m m r ir x y m m _r

u

e

z q

E

z e

q

r

r dr d

r

      









  

 

 

_{ }

 









2 , , 2 , sgn 4 (3) ₂ 2 , , 2

,

2

m n m n m m i i _m _m m _{k k} m n m n

k

E

z

e res G

e

       













      _    



 



   

  

(19)

INTEGRATION ALONG ENERGY DIRECTIONS

θ

Integration along segments





 

   



_{  }

_







   



_{  }

_







max min 1 2 1 2 (3) (3) (3) (3) 1 2 , , 1 2 2 2

0

1 0,

0, ,

,

, 0

2

2 ,

1 ,

,

m m m m x y m m ir ir m m m m m m m m

r

u

e

z q

E

z q

r

e

ir

e

ir

d

             



 



 



 

 









_

_





















_

_

_

_





















 

(20)

INTEGRATION ALONG ENERGY DIRECTIONS

Normal displacement comparison: Fast integration-classical integration

A

₀

mode

S

₀

mode

SH

₀

mode

Disc shaped transducer of radius a=5mm

Observation distance 100mm

Convolution

(21)

Normal displacement comparison: Fast integration-classical integration

A

₀

mode

S

₀

mode

SH

₀

mode

Convolution

Integration along energy directions

INTEGRATION ALONG ENERGY DIRECTIONS

Square shaped transducer of side L=9mm

Observation distance 100mm

(22)

Calculation of γ

m

(φ,z) and

excitability matrices E

m

(φ,z)

(SAFE+postprocessings)

and

saving for re-use

Determining φ and

R for the couple

source point

/ calculation point

Use of the matrix E

m

(φ,z)

for the calculated angle

Calculation of the Green’s

function g

m

(x,y,z)

Calculation of the field using

a summation of the different

source contributions

Loop over the source contributions:

1 loop for the integration thanks to Fraunhofer-like approximation

n loops for the integration along energy direction (one per ray direction)

P loops for the integration thanks to a convolution (one per point of the source