CC BY
25Electronic Journal «Technical Acoustics» http://www.ejta.org
2016, 5
H. Rhimini1*, M. El Allami2, M. Sidki3, A. Haddout1, M. Benhadou1
1 Laboratory of Industrial Management, Energy and Technology of plastics and composites materials. National High School of Electricity and Mechanics, Hassan II University, Casablanca, Morocco
2Laboratory of Mechanics Energy Electronics Telecommunications, Hassan I University, Settat, Morocco
3Department of Physics, Faculty of Sciences, Road Ben Maachou, BP20, 24000 El Jadida Morocco
Ultrasonic guided waves in tri-layer structure. Application to study the interaction of guided waves with hidden defect at low frequency
Received 25.06.2016, published 23.09.2016
The article presents a study of guided ultrasonic waves in a tri-layer structure formed of two aluminium plates assembled with a layer of epoxy resin. The resolution of the dispersion equation was used to plot the dispersion curves and to determine the energy distribution of a given mode propagating within the structure. On the other hand, the evaluation of the interaction of guided modes at low frequency with hidden defect placed either in one of the aluminium plates at the interface between aluminium and the adhesive layer was presented. The analyse of the reflected and transmitted waves in the dual space (wave number, frequency) permitted the separation and the identification of the propagating guided modes and showed the conversion phenomenon of the incident mode into other guided modes after the reflection and the transmission. Finally a quantification of the energy of the reflected and transmitted modes is presented.
Keywords: guided waves, adhesive structure, conversion modes, dispersion curves.
INTRODUCTION
In the industry of aeronautics, aerospace and automotive, the bonding replaces more and more riveting, bolting and wilding. The bonding offers many advantages over the traditional techniques. The principal interest is a high strength, a weight decrease, the possibility of linking various materials and thickness, facility to gather complex forms and provides a better distribution of stresses in the structure. During putting into service, the bonded structures can suffer damages caused by cyclic loading conditions [1] and adhesive bonded joint can be subject to various environmental conditions causing the degradation of the adhesive layer [2]. Thus, it is necessary to monitoring the state of the bonded assembly at various stages of its use. The ultrasonic non-destructive testing using the guided waves is very promising. Indeed, the guided waves are very sensitive to the change of size, position or mechanical properties of the various interfaces existing along their propagation. The use of guided waves as mean of monitoring the tri-layer structure requires the study and the understanding of the guided wave propagation in such structure.
"Corresponding author. E-mail : h.rhimini@ensem.ac.ma
Different studies have been carried out to explain the propagation of guided waves in multilayer structures. Thomson [3] and Haskell [4] are the first who studied the propagation of elastic waves in planar multilayered structures using the transfer matrix method. Lowe and Cawley [5] showed that Lamb waves are highly sensitive to mechanical properties and thicknesses of the adherents but remain insensitive to the properties of the adhesive layer. These results are similar to those obtained by Nagy and Adler [6]. Gao [7] proposed a modelling of the Lamb waves propagation in bonded plates by the finite element method. Heller [8] combined laser ultrasonic technique with the two-dimensional Fourier transformation (2D-FFT) to determine experimentally the dispersion curves of a tri-layer structure (two aluminum plates joined by an adhesive layer) and a two-layer structure (an adhesive layer joined to an aluminum plate). The obtained results show that the modes measured in the two structures are identical to those of a single aluminum plate. Seifried [9] combines the analytic model, the finite elements method and the experimental measures to develop a quantitative understanding of the propagation of guided Lamb waves in multilayered adhesive bonded components. Lindgren [10] studied the possibility of detection of defects in multilayer structures using high frequency guided ultrasonic waves. The results of their research have shown that the detection of defects is possible but at high frequencies the propagation of guided waves attenuate quickly if an adhesive material is present between the layers. Dalton [11] investigated the potential for long-range propagation of ultrasonic guided waves through double layered metallic structures using dispersion analysis and numerical modelling. Results have shown that to avoid attenuation over short distances, the guided waves should have frequencies bellow 1 MHz. Therefore it is important to identify and study low frequency guided ultrasonic wave modes that are insensitive to interface conditions, but offer good sensitivity for defect detection.
Our work presented here are aiming in first step to understand the propagation of guided waves in a tri-layer structure modelling two aluminum plates assembled by a layer of epoxy resin, by means of solving the equations of dispersion and determining the energy distribution of a given mode through the structure. Secondly we evaluate the interaction of the guided modes at low frequency with a notch modelling a hidden defect in the tri-layer structure. A characterization of this interaction is presented from the quantification of the energy of reflected and transmitted modes.
1. GUIDED WAVES: THEORY
1.1. Dispersion equation
The objective of this part is to establish the dispersion equation of tri-layer structures. Each layer is considered as isotropic, homogeneous and of infinite, with mass density , longitudinal velocities transversal velocity and layer thickness where indicates layer number 1 ,2 , 3 ) . In order to establish the dispersion equation, the scalar potential o and the vector potential y are used. The problem is supposed two dimensional (figure 1) so that the vector potential reduces to a scalar (y iz = y i).
Ay
d3--
d2-.
d,"
Free surface Layer 3: VL3,VT3,p3,h3 Layer 2: VL3,VT3,p3,h3 Layer!: VLl,VTl,p1,h1
Interface 2 Interface 1 x _
Free surface
Figure 1 : Geometry of the tri-layer structures
(1)
In the layer i two waves propagate:
- longitudinal wave described by the scalar potential o ;,
- shear wave described by the vectorial potential y ;. The expressions of o ; and are given by:
îo j = [Aid q i y + B ¡e-j q i y]ej (kx - ^, { = [C ¡ej sy + D ¡e -j sy]ej (kx- ^ ,
where A;, B ; , C ¡, D ; are the potential amplitudes in the layer i, k is the wave number parallel to the propagation direction, j is the imaginary number, o is the pulsation, q ; and s ; are the orthogonal wave numbers (real or complex), and satisfy dispersion
equations of guided waves k^ = k2 + qf and k| = k2 + sf.
The tangential and normal displacements in the layer derive from these potentials as follows:
(0
x dx öy'
u
(i) _ aoi a^i
y ay and the stresses are given by:
ax
= Hi (
y(i)
xy
a2o>i
a2Ti a2Ti
a2yj\ a y2 ),
(2)
(3)
(4)
(5)
dxdy dx2 dy2
where xj, ( are the Lamé coefficient in the layer i.
The potential amplitudes are determined from the following boundary conditions:
- First, consider the boundary condition at the upper surface (y=0), normal (oyy ) and tangential ( o^y ) stresses are zero at this position:
| -((xi + 2 j-) k2 + xi q2) (Ai + B ! ) - 2 Hi ks!( C ! - D i) = 0, { - 2 Hikqi (Ai - B i ) + (J-i( k 2 - s2 ) ( C i + D i) = 0 .
- Second, at a first interface (y = d ^ we consider continuity of stresses ( o^ = oyy,
(6)
and continuity of displacements (
(1)_„(2) 11(l)_ii(2)v
Ux - %
uy2 0 :
(7)
-((^ + 2 nO k2 + ^!q?)(Aiej1 1 d 1 + B ^ -i1 1d 1) - 2 ^ks-^C 1 es 1d 1 - D ^ -is 1d 1) + +((^ + 2 H2)k2 + ^2 q2)(a 2ei12 d2 + B 2 e -i 12d2) + 2 ^(C2^ s2d2 - D 2e -i s2d2) = 0,
-2 mkq^e 1 1 d 1 - B ^ -i 1 1 d 1) + 1( k 2 - s2)(C1 es id 1 + D ^ -i s id 1) + +2 n2kq2(A2e 12 dz - B 2 e -i d2) - ^ (k2 - s2 )( C 2eszdz + D 2e -i s2d2) = 0,
i k(A1 e 1 1d 1 + B 1e -i 1 1 d 1) + j s 1( C 1e s 1 d 1 - D 1e-i s 1d 1) --jk(A2 e 12d 2 + B 2 e -i 12d 2) js2( C 2e s 2d 2 - D 2e-i s2d2) = 0,
j q1(A1 e 1 1d 1 - B 1e-i 1 1 d 1) - j k( C 1e s 1d 1 + D 1 e -i s 1d 1) -- i q2(A2e^2 - B 2 e -i 12d 2) + jk(C 2es2d2 + D 2e -i s2d2) = 0 .
(2) (3)
Third, at the second interface ( ), we consider continuity of stresses ( and continuity of displacements ( :
-((^ + 2 H2)k2 + ^2 q2)(a 2ei 12 d2 + B 2 e -i 12d2) - 2 ^(C2^ s2d2 - D 2e -i s2d2) + +(a3 + 3 n3)k2 + A3 q|)(a 3eJ13 d3 + B 3e -i 13d3) + 2 ^ks 3(c3es3d3 - D 3e -i s3d3) = 0,
(8)
-2 n2kq2(A2e 12d 2 - B 2 e -i ^ d2) + n2 (k2 - sf)( C 2es2d2 + D 2e -i s2d2) + +2 n3kq 3(A3e 13 d3 - B 3 e -i 13 d3) - n3 (k2 - s | )( C 3eis3d3 + D 3e -is3d3) = 0,
j k(A2e^2 + B 2 e -i 12d 2) + is2(C 2es 2d 2 - D 2e-i s2d2) --j k(A3 e 13d3 + B 3 e -i 13d3) - j s 3( C 3e s3d3 - D 3e-i s3d3) = 0,
j q2(A2e 12d 2 - B 2 e -i 12d 2) - j k( C 2es 2d 2 + D 2 e -i s2d2) --j q 3(A3e ^3 - B 3 e -i 13d3) + j k(C 3es3d3 + D 3e -i s3d3) = 0 .
- Fourth, at the top surface (y = d 3 ) , the normal and tangential stresses vanish:
j -((A3 + 2 H3)k2 + A3 q2 )(A 3eJ 13 d3 + B 3e -i 13 d3) - 2 ^ks 3(C 3ds3d3 - D 3e -i s3d3) = 0 { - 2 n3kq 3(A 3ei^ d3 - B 3 e -i ^ d3) + n3(k2 - s |)(C3es3d3 + D 3e -i s3d3) = 0 . ( )
These equations resulting of continuity and boundary conditions, led to a homogeneous equations system characteristic of a propagation of the guided modes within the structure. This system can be written in matrix form:
[a](b] = {0}, (10)
where [ a] is a 12*12 matrix function of geometrical, mechanical properties of the structure, the excitation frequency and wave number , { } is a vector whose components are the 12 potential amplitudes.
The dispersion equation of guided wave is obtained by canceling the determinant of this system and solved numerically in order to determine the modes propagating in this structure.
In this study, we consider tri-layer structure composed with two aluminum sheets bonded by an adhesive layer of epoxy resin. The properties are similar to the specimen examined in [9]. The thickness of each aluminum plate is 0.9398 mm, while the adhesive layer is 0.25 mm thick.
Longitudinal and shear velocities in aluminum plate are respectively VL = 6 1 5 0 m /s and VT = 3 1 0 0 m/s and its density is p = 2 700 kg/m 3 , while longitudinal and shear velocities in adhesive layer are taken respectively VL = 7 7 1 m/s and VT = 3 70 m/s and its density is p = 1 1 0 6 kg/ m 3 .
1.2. Dispersion curves
To determine the propagating guided modes in the structure, we must find all the pairs ( f, k) solution of the dispersion equation obtained by canceling the determinant. In this context we have developed a program to calculate the dispersion curves of any isotropic tri-layer structures. Zeros of the dispersion equations are determined and the modes are then distinguished and separated. Indeed, for every pair ( f,Vp) we determine the zeros of the dispersion equation by the bisection method. These couples zeros of the dispersion equation are stored in a matrix and an algorithm of separation of modes is then applied. We then obtains dispersion curves frequency f as a function of the phase velocity Vp. From these curves are deduced other representations such as wave number versus frequency using the formula k = o /Vp and group velocity as a function of frequency using the following equation: Vg = d o /a k .
Figure 2 shows the phase velocity dispersion curves in terms of frequency for different modes in tri-layer adhesive joint with material properties as described in section 1. The modes are identified with M and number. To investigate this curve some modes appear only after a certain frequency (cutoff frequency). For example in the frequency intervals of 0 kHz to 500 kHz only four modes of M1, M2, M3 and M4 propagate, and other modes don't propagate in this low frequency. M1 and M2 appear in 0 MHz, while M3 and M4 appear respectively in 105 kHz and 225 kHz (see figure 3).
Figure 2: Phase velocity dispersion curves for tri-layer adhesive joint: Aluminum/Epoxy/Aluminum
Figure 3: Wave number dispersion curves for tri-layer adhesive joint: Aluminum/Epoxy/Aluminum
1.3. Displacements field
The expressions of the displacements field iix and iiy through the structure are given by the following relationships:
fii® = [/ Zc(i4¿e^ ^ +5e-i? + ^¿(Qe^ - (fcx -ÛJ
(u® = [/^(4 ¿e^ ^ -^¿e-^-yfc(Qe^ + e-^)]eJ-ÙJ
(11)
— I ifl-l A .pJHiy — R.p-JHiy I — iW I'.pJ^iy A- D.p-J^iy II
y
To determine the constants 4¿, Q and D£, we proceed as following. We first build a linear system of the form [a ' ]{b'} = {c} deducted from the previous system [a]{b } = {0}. Where [ a ' ] is a square matrix formed of the first eleven rows and first eleven columns of the matrix [ a], { b'} is a vector consisting of the first eleven components of the vector { b } and { c) is a vector containing the opposite of the first eleven components of the twelfth column of the mtrice [ a]. Then by triangularization of the matrix [ a '] we obtain the values of the eleven constants based on the twelfth constant Thus, the displacements distribution thought the structure are then known to a constant.
1.4. Stress field
Using Hooke and strain-displacement relations, the stresses in the tri-layer structure are written as follows:
^ = [- (Uj + 2h£) k2 + 4<7t2) (4 ¿e^ tf + tye"-^) - Qe^- + De"^-)] wf), o-W = [- ((4 + 2^) q2 + Aik2) (4 ¿e^ ^ + fi.e"^-) + 2^( Qe^ + De"^-)] e^(to" wf), (12) = [- 2^/^(4 ¿e^- - fi£e" + ^ (k2 - s2)(qe^- + De" toy)]e'(k*" wf).
1.5. Guided wave energy
The acoustic power carried out by a mode along direction through a cross-section localized in plane with length in the direction and thickness of which along axis is /, is the flux of Poynting vector P [12]:
Pac = ^^Pn dy dz, (13)
where P is the normal to the section defined by .
is a complex number, whose part is denoted corresponding to the temporal average carried by the mode M through the tri-layer structure,
rh,
H e I
2
^m = - ^ e J^VM. ^M dy , (14)
where v is the velocity vector and <o is the stress tensor of the mode M. * design the complex conjugate. In the plane oxy vM = [i^Mv*M] and <oM = {j^}
1.6. Normalized displacements
To determine the energy distribution of the incident mode between the generated modes in the structures, these modes must be normalized in power flux. The expression of the normalized displacements and are now calculated as follows:
Mr
= and ^ - = (15)
1.7. Power coefficients
After interaction of the incident guided mode with defect, there is apparition of a finite number of M guided modes in the tri-layer structure, before and after the defect. The reflected coefficient is defined by the ratio between the energy of the reflected mode qf^ and the energy of the incident mode <qin c: R = q^/qtnc. Similarly, the transmitted coefficient is defined by: T = qjj ans/qinc, where qf^ans is the transmitted mode energy. qinc, q^ and qf^ans are defined as follows:
/ a inc \z , /Aref\2 / ,trans\^
= (V), < = (#■), <ans = (V), (16)
pine
where Ain c, A „e/ and A % ans are amplitudes respectively of the incident, reflected and transmitted modes. Those amplitudes are determined after processing applied displacements of guided modes signals in the studied structure.
2. MODELLING OF GUIDED WAVES PROPAGATION: THE FINITE ELEMENT METHOD
The discretization of a tri-layer structure and the application of the virtual theorem allow writing the motion equation in the following matrix form:
[M]{ U} + [ff]{ U) = {F), (17)
where [M ] is the global mass matrix, [ fr] is the global stiffness matrix, { //} is the displacement vector, { U} is the acceleration vector and { F) is the vector of applied forces. The damping is not considered in this study.
To solve the equation (17) and find the displacement field { U), we use the Newmark method. The construction of the solution at time t + A t is done from vectors at time t: { Ut), { t/t} and { U'J according to the following algorithm [13]:
({ Ut+a J = { UJ + A t (( 1 - a) { UJ + a{ U t+A t} ) >
{ //1 X \ (18)
({Ut+a t) = { Ut) + A t{ Ut} + A t2 (g - b) { Ut} + b{ Ut+a t}) -
where a and b are Newmark integration parameters, At is the time step.
The time step At and the spatial discretization steps are crucial parameters for the convergence of the finite element method. For the modelling of the guided wave propagation, a two-dimensional FEM models are used, in which four-node plane strain elements are employed for a tri-layer structure were developed and dynamic simulations were accomplished using the commercially available software Abaqus/Explicit based on an explicit dynamic Finite Element Method. For good convergence, the smallest propagating wave length in the ultrasonic pulse mast be discretized with at least 20 nodes and the integration step times At must be less than (1/20fmax), where fmax is the bigger frequency exist in the ultrasonic signal [14].
3. POST PROCESSING OF DISPLACEMENT FIELDS
To identify and visualize the contents modes in temporal and spatial frequency in the structure, a double Fourier Transform will be applied to the normal displacement fields taken at the upper interface of the tri-layer structure. The bi-dimensional Fourier transform of the spacetime of displacement u (x, t) is defined by the formula
Fu (6, k ) = J_+J u (x, t) e j(fcx - w ^d x d t, (19)
where < is the angular frequency and j is the complex number such as j2 = — 1.
4. NUMERICAL SIMULATION
4.1. Description of the studied sample
The studied sample is a tri-layer structure was made of two 3 mm thick aluminum plates with a width of 70 mm and a length 500 mm bonded with an approximately 0.25 mm thick epoxy. The acoustic properties of the structure are similar to those defined in section 1. We compute reflected and transmitted power coefficient for a tri-layer structure contains a hidden defect. The defect is simulated with a rectangular notch. The notch was machined either in of the aluminum plates at the interface between aluminum layer and the adhesive layer a height (p=0.5 mm) and width (w=0.8 mm). The notch was placed in the middle of the structure.
Figure 4: Sample studied
y
4.2. Generation of guided modes
To generate the mode M in the structure and study its interaction with the considered defect, we apply on the left edge of the tri-layer structure (x=0, y) the analytical displacements (equations 11) normalized by the power flow through the structure thickness (figures 5a, 5b, 5c and 5d). The spatial distribution of the displacements is applied during 10 cycles tone burst weighted by a Hanning window centered on the excitation frequency (figure 5e). The selected excitation frequency is f=200 kHz (figure 5f). For this value, the wave numbers of the modes M i,M2, M 3 and M 4 are respectively k M 1 = 62 8 m " 1, K M 2 = 5 65 m " 1 , KM 3 = 2 3 4 m " 1 , KM4 = 2 2 6 m _1.
a i i XT --— u* — Uy
; l\
I \
/ t
1 l \J
I
b I 1 L i —-f^SH ---Uy
1
. -----
-----1—— -
1 / i ! i i i
I ^ : \ !
c { ---Ux -Uy
-------
/
Normalized displacements (>jti)
Normalized displacements (vim)
Normalized displacements (jim)
-2 -1 0 1 Normalized displacements Cum)
I 400
M, / /
/ Hj if*'>
Frequency f (MHz)
Figure 5: Normalized displacements applied to the left edge of the tri-layer structure at f=200 kHz to generate: Mt mode (a), M2 mode (b), M3 mode (c) and M4 mode (d). Time profile of the excitation (e) and the selected frequency 200 kHz (f). Grey area - adhesive layer
f
d
e
4.3. Displacements of M1? M2, M3 and M4 modes
Two series of 200 nodes are monitored either along the upper surface of the structure. These considered nodes are regularly spaced of 0.5 mm and taken far from structure edge and defect to avoid non propagative and vanishing modes and thus to consider only the propagative modes. Nodes located before the defect control incident and reflected modes while those located after the notch control the transmitted modes. The normal displacements depicted before and after the defect are represented in the space-time plane when guided modes Mi, M2, M3 and M4 are excited (see figure 6, 7, 8 and 9). On these figures, we can see clearly the wave packets: incident reflected and transmitted one. The parallelism of ridges in the incident wave packet permits to get phase and group velocities of guided mode M [15]. In the figure 8a, the ridges of the incident packet wave mode are not parallel. So, this wave packet propagates with different phase and group velocities. This shows that another incident wave is excited in the structure with a lower amplitude. In the reflected and transmitted wave packet (figures 8 and 9), the ridges are as not parallel showing that either at the reflection and transmission, the guided mode M is converted in other guided modes. The signals present in Figures 6b and 7b show that the amplitudes of the reflected modes are too low compared to the incident modes. On the other hand, the amplitudes of transmitted modes are considerable (figures 6d and 7d). Which shows that a large part of the energy of the incident wave is transmitted through the defect when M1 and M2 are excited at f = 200 kHz. Figures 8b and 9b show that the amplitudes of the reflected modes are important. So the interactions of the modes M3 and M4 are good.
100 120 140 Postton before defect (mm)
-1 -0.5 0 0.5 1 Amplitude (urn)
320 340 360 3S0 Position after defect (mm)
1 -0.5 0 0.5 Amplitude (um)
Figure 6: Time evolution of displacements of monitoring zones before (a) and after (c) the defect and at two points before (x=100 mm) (b) and after (x=340 mm) (d) the defect when M1
is incident at f=200 kHz
d
b
c
a
220 200 175 150
§ 1 100
1
fiS?»^»..- -
---- - ™ ™
""1
220 200 176 160
I"5 1 100
—b-
100 120 140 Position before defect
-Ф-
-2-1 0 1 2 Amplitude (pm)
340 360 380 n after defect (mm)
1 -0.5 0 0.5 1 Amplitude (jim)
d
b
c
a
Figure 7: Time evolution of displacements of monitoring zones before (a) and after (c) the defect and at two points before (x=100 mm) (b) and after (x=340 mm) (d) the defect when M2
is incident at f=200 kHz
Figure 8: Time evolution of displacements of monitoring zones before (a) and after (c) the defect and at two points before (x=100 mm) (b) and after (x=340 mm) (d) the defect when M3
is incident at f=200 kHz
100 120 140 160 Position before defect (mm)
-0.5-0.25 0 0.25 0.5 Amplitude (цт)
300 320 340 360 380 400 Position after defect (mm)
.5-0.25 0 0.25 0.5 Amplitude (um)
d
b
c
a
Figure 9: Time evolution of displacements of monitoring zones before (a) and after (c) the defect and at two points before (x=100 mm) (b) and after (x=340 mm) (d) the defect when M4
is incident at f=200 kHz
4.4. 2DFFT post-processing
To identify the existing modes in the structure after the reflection and transmission, we applied a classic double Fourier transform (2DFFT) [17] to normal displacements, depicted before and after the defect, to get its spectrum in the (k, f) space. Figures 10, 11, 12 and 13 show a contour plot representation of the obtained spectrum. An artificial superposition of the exact dispersion curves of the tri-layer structure (solutions of the dispersion equations (10)) to this spectrum is done to identify the reflected modes (k<0) and the transmitted modes (k>0).
Figures 10a and 10c (respectively 11a and 11c), illustrate the contour plot spectrum of the normal displacements on the upper interface of the tri-layer structure when the incident mode is M1 (respectively M2). These figures show clearly that the reflected and transmitted modes are only the incident mode. A cut of the representation (f, k) performed at f = 200 kHz, is shown in Figures 10b and 10d (respectively 11b and 11d). This representation is used to determine the amplitudes of the incident, reflected and transmitted modes when M1 is incident (respectively M2) [16]. Figure 12a illustrates the contour plot spectrum of the normal displacements on the upper interface of the tri-layer structure when the incident mode is M3. On this figure, another incident wave can be observed in the tri-layer structure a lower amplitude. But at f=200 kHz, only one incident mode must be in incident packet wave. In the figure 12b, we can identify the parasitic incident wave: it is the M2. Figure 12a and 12c show that in this precise case, six modes propagate in the structure, which are: The two incidents modes M3 and M2 (parasitic incident mode), two reflected modes M2 and M3 and two transmitted modes M2 and M3.Figures 13a and 13c, show the contour plot spectrum of the normal displacements on the upper interface of the tri-layer structure when the incident mode is M4. These figures show clearly that after reflection and transmission on the defect, the incident mode M4 is converted to M2 and M4 modes at the selected frequency 200 kHz. The spectral magnitudes of all these modes are illustrated on figures 12b and 12d (respectively 13b and 13 d) for comparison.
These representations have allowed us to identify the propagating modes in the tri-layer structure containing a defect when a mode M is excited at the frequency f = 200 kHz and allow to quantify energy of reflected and transmitted modes.
4.5. Reflection and transmission coefficients
In order to determine the energy distribution of incident mode (M1, M2, M3 or M4) between the reflected and transmitted modes, it is necessary to quantify the amplitudes of each mode propagating in the structure. The peaks values obtained in Figures 10b, 10d, 11b, 11d, 12b, 12d 13b and 13d are proportional to modes amplitudes. The coefficient of proportionality is the same for all of the modes. From these representations (figures 10b - 13b) (respectively 10d, -13d), we get, at positions 200 kHz, the amplitudes of the incident and reflected modes, and respectively the transmitted modes. The displacements are normalized in power and the theoretical coefficient, relating the amplitude of surface displacement to Poynting vector, is used to obtain the energy percentage transported by each wave. The reflection and transmission coefficients are computed by referring to relation (15).
0.2 0.3 04 0.5
Frequency f (MHz)
Amplitude
Frequency f (MHz)
Amplitude
Figure 10: Superimpostion of analytical dispersion curves to the energy repartition in the dual space (k,f) obtained by the 2D-FFT processing of displacements. Incident M1 at f=200 kHz
1200 1000
-1000 -1200,
Frequency f (MHz)
0.2 0.3 0.4 Frequency f (MHz)
0 0.2 0.4 Amplitude
Figure 11: Superimpostion of analytical dispersion curves to the energy repartition in the dual space (k,f) obtained by the 2D-FFT processing of displacements. Incident M2 at f=200 kHz
12001 1000
0.2 0.3 0,4
Frequency F (MHz)
tscjji- ey ■ (Ml '2i
0.02 004 Amplitude
Figure 12: Superimpostion of analytical dispersion curves to the energy repartition in the dual space (k,f) obtained by the 2D-FFT processing of displacements. Incident M3 at f=200 kHz
02 0 3 0-4 Frequency f (MHz)
1200 1000
02 0.3 0.4 Frequency f (MHz)
0 0.05 0.1
Amplitude
Figure 13: Superimpostion of analytical dispersion curves to the energy repartition in the dual space (k,f) obtained by the 2D-FFT processing of displacements. Incident M4 at f=200 kHz
d
b
c
a
d
b
c
a
d
b
c
a
d
b
c
a
In table 1 are reported the reflection and transmission coefficients values of each incident mode and the corresponding reflected and transmitted mode. From these values we can make the following remarks:
- When the guided mode M1 is incident the relative error in the energy balance is equal to 0.63% and the reflected mode energy remains lower than 1%. The interaction of the guided mode M1 with the defect is weak. It cannot used to characterize the defect.
- When the incident guided mode is M2, the relative error in the energy balance is higher than 2% and the reflected mode energy is almost equal to 2%. The interaction of the guided mode M2 with the defect is medium.
- When the guided mode M3 is incident the relative error in the energy balance is higher than 3% and the reflected mode energy is almost equal to 60%. The interaction of the guided mode M3 with the defect is good, but the major disadvantage is that during excitation another parasitic mode generates simultaneously with the M3 mode. This mode cannot be chosen to characterize this defect.
- When the guided mode M4 is incident the relative error in the energy balance is equal to 0.75% and the reflected mode energy remains higher than 1%. The interaction of the guided mode M4 with the defect is medium. This mode is suitable for the characterization of the defect.
Table 1 : Energy balance
Incident wave Reflected wave Transmitted wave kinc(m-1) kre/m"1) ktrans(m ) R T R+T Relative error (%)
M1 M1 M1 628 -628 628 0.0091 0.9972 1.0063 0.63
M2 M2 M2 565 -565 565 0.0193 0.9658 0.9761 2.39
M3 M3 M2 M3 M2 235 565 -235 -565 235 565 0.5976 0.00003 0.4354 0.0001 1.0331 3.31
M4 M2 M4 M2 M4 226 -565 -226 565 226 0.0041 0.0071 0.0012 0.9951 1.0075 0.75
kinc , incident wave number; kref, reflected wave number ktrans , transmitted wave number, R, reflection coefficient and T, transmitted coefficient
5. CONCLUSION
In this paper, we proposed a method for solving guided waves dispersion equations in a tri-layer structure, and have determined the distribution of the energy of a given mode through the structure. On the other hand, we studied the interaction of the guided modes Mi, M2, M3 and M4 with a hidden defect in a tri-layer structure. The reflected and the transmitted waves by the defect were separated and identified in the dual plane (wave number, frequency). The quantification of the energy of reflected and transmitted modes showed that the M4 mode is the most appropriate for characterization of the hidden defect. The results presented in this paper are also useful to set and optimize the experimental Non-Destructive Testing (NDT) technique of hidden defect detection in the tri-layer structure.
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