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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
Background and objectives
General field expression
Finite-sized sources
Summary
BACKGROUND AND OBJECTIVES
•
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)
Phase velocity (km/s)
BACKGROUND: Guided waves non destructive testing of plate-like
structures
Phase velocity dispertion curve
e = thickness 20 mm
f = frequency 0,2 MHz
e x f [MHz.mm]
Target applications:
•
•
•
•
Non-Destructive Evaluation
Structural Health Monitoring
Acoustical Emission
…
In all cases transducers of finite size
are used
3/24
OBJECTIVES: An efficient model to predict guided wave radiation by
finite-sized sources
•
•
•
•
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
THE BASES:
•
•
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
Idea: try to combine SAFE and Fraunhofer-like approximation
[1] Taupin, Lhémery and Inquité, J. Phys. : Conf. Ser., 269, 012002 (2011).
[2] Raghavan and Cesnik, Smart Mater. Struct., 14, 1448 (2005).
4/24
Summary
GENERAL FIELD EXPRESSION
• Convolution of the source and a Green’s function
u(3)  x, y, z, q (3)    g (3)  x  x ', y  y ', z  q(3)  x ', y '  dx ' dy '
S
•
q
With :
u(3) , displacement
g(3) , Green’s function
q(3) , source
•
Modal Green’s function :
g (3)  x, y, z   

g (3)
m , n  x, y , z 
m n ( m, )
n is the number of phase contributions for a given observation direction and a given mode m
6/24
SH0 mode
[0°/90°]S T700GC/M21 cross-ply composite fiber-reinforced polymer
f = 300kHz
for this energy (observation) direction three phase contributions
7/24
FAR FROM CAUSTICS
phase approximation thanks to three parabolas (second order approximation)
and then sum of the three contributions
8/24
• Expression of the modal contribution far from caustics
•
g
(3)
m,n
Calculated thanks to stationary phase method [1]
 x, y , z   
x y
2
2

1/2
e
i x 2  y 2  m ,n   m ,n , 
e
  2  m ,n  m ,n ,  

 
i sgn 
2

 i
4



 2
km , n   m , n 
e res G k  k  
m ,n m ,n
2
With tan   


y
,
x
  
 2  m,n   m,n ,  
 2

 m ,n  m ,n ,  k m ,n  m ,n cos  m ,n  phase term,
G spatial Fourier transform of g,
k m ,n wavenumber calculates thanks to the SAFE method
[1] Velichko and Wilcox, J. Acoust. Soc. Am., 121, 60 (2007).
9/24
NEAR CAUSTICS
phase approximations:
- parabolic approximation
- cubic approximation (third order approximation)
10/24
• Expression of the modal contribution near caustics
•
Calculated thanks to stationary phase method [1]
g m(3),1 2  x, y, z  

x2  y 2

1/3
i
ei
x 2  y 2 L  
Ai  



  res G k  k   km,n   m,1 
m ,1 m ,1




e2
2
2 S   m,1 
 2 m,n   m,1 ,  


2 S   m,2  
 2 m,n   m ,2 ,   

2



 res G k  k   km,2   m,2 
m ,2 m ,2
 2
With
 



1
L   m ,1  m ,1 , m ,2  m ,2 ,  , S     3  
2
 4  m, 2  m, 2 ,    m,1  m,1, 
2
   x 2  y 2 3 S   , and Ai Airy function



2
 3 ,


[1] Karmazin, Kirillova, Seeman and Syromyatnikov, Ultrasonics , 53, 283 (2013).
11/24
Summary
FINITE-SIZED SOURCES
ISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION
L
X
(xs,ys)
LY
(xs,ys)
a
Simplification of the displacement field expression, thanks to the isotropic
properties of the material:
u
(3)
 x, y , z , q    e
(3)
m

S
i

4
res G k  k

 x  x '   y  y '
2
e

 x  x '   y  y '
2
2
ikm
 
1/2
m
km (3)
q  xs , ys 
2
 x  x ' 2   y  y '2
2
ikm
e
 x  xs    y  ys 
2

2
1/2
e
ikm
 x  x '2   y  y '2

2
2
x

x

y

y
 s   s  1



dx ' dy '

 x  xs 
 y  ys 
x

x
'

y

y
'



 
2
2
2
2

 x  xs   y  ys 
 x  xs   y  ys 

1/2
13/24
ISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION
LX
(xs,ys)
LY
(xs,ys)
a
u (3)  x, y, z, q (3) 


 x  xs    y  ys 
2

Fm,rect  x, y   LX LY sinc 


Fm,disc  x, y    a 2
2

1/2
e
i
 x  xs 2  y  ys 2 km  m 
k  km   m 
m
km   m  x  xs 
 x  xs    y  ys 
2
km   m 
Fm, fraun  x, y  q (3)  xs , ys 
2

e 4 res G 
i
2


LX 
sinc 

2 


km   m  y  ys 
 x  xs    y  ys 
2
2

LY 
2 

2 J1  k m   m  a 
km   m  a
14/24
ANISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION
•
y
x
Plate characteristic
•
•
[0°/90°]S T700GC/M21 cross-ply composite
fiber-reinforced polymer
Plate thickness: 1mm
C11
(GPa)
C22=C33
(GPa)
C12=C13
(GPa)
C23
(GPa)
C44
(GPa)
C55=C66
(GPa)
Mass
density
(kg/m3)
Ply
thickness
(mm)
123.4
11.5
5.6
6.4
2.6
4.5
1.6x103
0.25
f = 300kHz

15/24
Observation points
Finite-sized source
Computed results:
uz  R  100mm,  
16/24
ANISOTROPIC PLATE: FRAUNHOFER-LIKE APPROXIMATION
Normal displacement comparison
Disc shaped transducer of radius a=5mm
Observation distance100mm
A0 mode
Convolution
Fraunhofer-like approximation
Even for the less anisotropic
mode: approximation fails
17/24
INTEGRATION ALONG ENERGY (OBSERVATION) DIRECTIONS
θ
Integration along segments (change
of variables in surface integral)
u (3)  ex  0, ey  0, z, q (3)   
max r2  
 
m  min r1  
(3)
m
E
 , z   e
i

2
res G k  k
km   m 
m
 m 
  m,n   m,n ,  
  2  m ,n   m ,n ,  

i sgn 
2


4





e
2
2
      
1 (3)
ir '   ,
Em  , z  e m  m   q (3)  r ',   r ' dr ' d
r'
 2
18/24
INTEGRATION ALONG ENERGY DIRECTIONS
θ
Integration along segments
u
(3)
e
x
 0, ey  0, z , q
 max
(3)
 
m  min
 r  0   r2  0  
Em(3)  , z  q (3)  1
,0
2
 r1    r2   




2


1
 eir1  m  m ,   ir1    m   m ,    1 eir2  m  m ,   ir2    m   m ,    1 


 d
2
2


,



,







m
m

m
m

19/24
Disc shaped transducer of radius a=5mm
Observation distance 100mm
Normal displacement comparison: Fast integration-classical integration
A0 mode
S0 mode
SH0 mode
Convolution
Integration along energy directions
20/24
Square shaped transducer of side L=9mm
Observation distance 100mm
Normal displacement comparison: Fast integration-classical integration
A0 mode
S0 mode
SH0 mode
Convolution
Integration along energy directions
21/24
Flow chart
Calculation of γm(φ,z) and
excitability matrices Em(φ,z)
(SAFE+postprocessings)
and saving for re-use
For a calculation point (x,y), for a given mode
Calculation of the field using
a summation of the different
source contributions
Determining φ and
R for the couple
source point
/ calculation point
Use of the matrix Em(φ,z)
for the calculated angle
Calculation of the Green’s
function gm(x,y,z)
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)
22/24
Summary
SUMMARY
Summary
SUMMARY
•
• Fraunhofer approximation OK for isotropic plates
• Similar approximation fails for anisotropic ones
•
Development of a modal integration method over finite-sized sources
• Deals with arbitrary multilayered anisotropic plates (SAFE for modes)
• Integration over energy directions
• Faster than a classical surface integration
• Takes into account the field near caustics
•
Developed for:
• normal stress sources (as shown here)
• tangential stress (similar expressions)
24/24
Commissariat à l’énergie atomique et aux énergies alternatives
Centre de Saclay | 91191 Gif-sur-Yvette Cedex
Etablissement public à caractère industriel et commercial | R.C.S Paris B 775 685 019

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