Interaction of Lamb modes with an inclusion

Ultrasonics 53 (2013) 130–140

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Interaction of Lamb modes with an inclusion G. Shkerdin 1, C. Glorieux ⇑ Laboratorium voor Akoestiek en Thermische Fysica, Departement Natuurkunde, K.U. Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium

a r t i c l e

i n f o

Article history: Received 16 January 2012 Received in revised form 29 April 2012 Accepted 30 April 2012 Available online 1 June 2012 Keywords: Guided acoustic waves Lamb waves Multilayer Wave–defect interacton Non-destructive testing

a b s t r a c t The interaction of Lamb modes propagating in a steel plate containing a thin inclusion is analyzed for cases where the inclusion material has elastic parameters similar to the ones of the plate, and where the inclusion is in perfect mechanical contact with the surrounding plate material. A modal decomposition method is used to calculate the conversion of an incident Lamb mode to other modes. Hence, the influence of the type of incident mode, of the location and geometry of the inclusion, and of the elastic parameters of the inclusion and plate material on the mode conversion coefficients is analyzed. Besides the expected increase of the conversion efficiency with increasing cross section of the inclusion, it is found that due to reasons of symmetry, the presence of an inclusion leads to an efficient conversion of an incident S0 mode into reflected and transmitted A0 modes, unless the inclusion is located very close to the plate center. On the other hand, the conversion efficiency of an incident A0 mode into a reflected A0 mode is found to be strongly dependent on the depth of the inclusion, this conversion even disappearing for some location depths. For the cases studied, the inclusion location dependence of the mode conversion seems to be correlated with the normal profile of the longitudinal normal stress component ryy ðyÞ. As intuitively expected, the mode conversion efficiency increases with the mismatch of an acoustic impedance like factor between the uniform plate and the inclusion region. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Motivated by applications for non-destructive characterization of materials, the interaction of acoustic waves with defects inside materials has been widely studied during the past years (see, e.g., Refs. [1–9]). Among the wide variety of defects in plate-like materials, inclusions, i.e. regions inside a plate with density and elastic properties different from these of the plate material, represent an important class. For example, cracks and delaminations filled with air or dirt can be considered as inclusions, and their timely detection by means of non-destructive testing techniques can be very crucial for reasons of safety or performance in a production line or transport container. Exciting and detecting Lamb modes offers a quite convenient way to remotely monitor the state of large regions in a plate of interest (e.g. the wall of a container or pipeline, the tail boom of a helicopter, or the walls of an airplane cabin), since Lamb waves easily scatter on acoustic heterogeneities. The interaction of Lamb modes with different kinds of delaminations (see, e.g., Refs. [1–5]) and notches or holes in plates (see, e.g., Refs. [6–9]) has therefore attracted quite some attention.

⇑ Corresponding author. E-mail address: [email protected] (C. Glorieux). Address: Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Vvedensky sq., 1, Fryazino, Moscow Region, Russia. 1

0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2012.04.008

Due to their wavelength typically being of the order of the plate thickness, Lamb modes are very dispersive and the possible number of propagating symmetric and antisymmetric modes strongly depends on frequency. When a Lamb mode encounters a heterogeneity, then part of its energy is channelling to reflected and transmitted modes of different types. Till now, the interaction of Lamb modes with defects was mostly analyzed for an acoustically thin plate, in which case only the lowest order symmetric and antisymmetric Lamb modes can propagate. The existence of only a few modes in a detected displacement signal allows to easily discriminate between different (defect-independent) excited modes on one hand, and defect-induced modes on the other hand. The propagation of Lamb modes through a thin absorptive bi-layer with delamination was theoretically analyzed in Ref. [10], where it was shown that, although the incident energy flow conversion into reflected and another transmitted modes is very small, delaminations can be experimentally assessed analyzing the tangential dependence of the normal displacement along bi-layer surfaces. The amplitude of the normal displacement at the plate interfaces is frequently measured to analyze propagating acoustic waves (see, e.g., Refs. [11,12]). Notches or holes filled with air or soft dirt represent a subset of inclusions for which the difference between the inclusion and plate material parameters is very large. As a result, the acoustic wave diffraction on these defects is typically quite strong, leading to a considerable fraction of incident energy being reflected, and to

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significant mode conversion in transmission. As a result, notches and holes can be quite easily detected by analyzing the modal content in reflected or transmitted displacement signals detected along the plate of interest. Quite often though, inclusions are very thin and the elastic parameters of the material filling an inclusion can be quite similar to the ones of the surrounding material, resulting in small and thus difficult to detect influences on the wave propagation. The present paper is concerned with the latter situation and aims at quantifying the mode conversion coefficients in the case of weak interaction, and at gaining insight in the factors determining the sensitivity of different types of Lamb modes to the geometrical and elastic characteristics of the inclusion and its surroundings. Although our interest here is mainly oriented towards the detection limit, in terms of trends, aspects of symmetry and order of magnitude of effects of inclusions on the signal, the results can easily be extrapolated to defects with higher contrast. The calculation is based on a modal decomposition method that was introduced earlier for monolithic plates and for bi-layers (see Refs. [13,14]). The generalization of the method to the case Lamb mode interaction with an inclusion in a plate is concisely summarized in Section 2. In Section 3, the model is applied to the systematic study of the dependence of the conversion of an incident mode on the parameters of the inclusion. The mode conversion is assumed to be monitored via the normal displacement amplitude at the plate surfaces. In this part, also the effect on the numerical calculation accuracy of truncating the infinite system of equations arising from implementing boundary conditions on the infinite set of modes is analyzed.

The components of the stress tensor Snik and the mechanical displacement vector U ni ði ¼ y; z and k ¼ y; zÞ of every (nth) right-going Lamb mode are given by:



        Snik ; U ni ¼ rnik ; uni Expðiqn z  ixtÞ þ rnik ; uni Exp ixt  iqn z

where the asterisk symbol refers to complex conjugation and the functions rnik ðyÞ and uni ðyÞ depend on y, the coordinate normal to the plate, which is taken to be zero at the bottom of the plate. Every Lamb mode wave number qn is satisfying the Lamb dispersion equation (see Refs. [14,15] for the monolithic plate region and see Appendix A for the threefold composite inclusion region 0 < z < LÞ. The normal depth profiles (y-direction) of the dynamic variables Snik ; U ni ; rnik ; uni of the different modes (index n) are all different, but every mode satisfies the wave equation, the stress–strain relations and the stress-free conditions at y ¼ 0 and y ¼ d. The Lamb modes in the composite inclusion region ð0 < z < L, see Fig. 1) satisfy additional continuity conditions reflecting perfect mechanical contact at the tangential interfaces y ¼ d1 ; d2 . All Lamb modes are mutually orthogonal (see e.g. Eq. (2) in Ref. [10]). The expansion of the mechanical displacement vector for the transmitted acoustic waves ~ ur ðz > LÞ can be written as:

!tr X tr n U ¼ Cn ~ utr Expðiðqn ðz  LÞ  xtÞÞ

ð2Þ

n

with C trn the expansion coefficients for transmitted acoustic waves, which are to be determined. The expression is analogous for the reflected acoustic waves ðz < 0Þ:

!ref X ref n U ¼ Cn ~ uref Expðiðqn z þ xtÞÞ

ð3Þ

n

2. Modal decomposition method in a plate with an inclusion The two-dimensional geometry (we consider cases where there are only displacement and stress components in the YZ plane) under consideration is shown schematically in Fig. 1. An incident right-going sinusoidal Lamb wave with angular frequency x and known magnitude propagates along the z-direction in an isotropic plate with thickness d. The plate surfaces at y ¼ 0; y ¼ d are everywhere stress-free. A rectangular inclusion, located in the intersection between the regions d1 < y < d2 and 0 < z < L, is assumed to be in perfect mechanical contact with the surrounding plate material, resulting in permanent continuity of the mechanical displacement components and of the proper stress tensor components at the interfaces y ¼ d1 ; y ¼ d2 for 0 6 z 6 L and the interfaces z ¼ 0; z ¼ L for d1 6 y 6 d2 . The method to calculate the acoustic field resulting from the incident wave, and from converted waves that are generated due to its interaction with the inclusion, is based on an expansion of this field in the Lamb eigenmodes in each region of the composite plate, and on the use of boundary conditions at the vertical planes z ¼ 0 and z ¼ L to determine the expansion coefficients. Since this method is described in detail in Refs. [10,13] for the case of delaminations, here we just give the concise description of its generalization for an inclusion.

ð1Þ

The expansion of the mechanical displacement vector for rightgoing acoustic waves in the region 0 < z < L (indicated by index ‘1’) ~tr;1 , is the same as the one of in Eq. (2), containing the inclusion, U ~n ~n provided the following substitutions: C trn ! C tr;1 n ; utr ! utr;1 ; qn ! qn;1 and L ! 0. The expansion for the left-going acoustic waves ~ref ;1 in this region is given by Eq. (3), with analogous substitutions U and z ! z  L. Applying the boundary conditions expressing perfect mechanical contact at the normal interfaces z ¼ 0; z ¼ L and using the orthogonality condition as in Ref. [10] an infinite system of linear equations Eqs. (4)–(7) in the unknown mode coefficients ;1 C tr;ref ; C tr;1 and C ref can be derived: n n n

AðnÞC trn ¼

X

;1 iqm;1 L C tr;1 ðK n;m þ In;m Þ þ C ref m e m ðI n;m  K n;m Þ



ð4Þ

m

AðnÞC ref n

 X ref ;1 ¼ C m eiqm;1 L ðK n;m þ In;m Þ þ C tr;1 m ðI n;m  K n;m Þ m

A1 ðmÞC tr;1 m ¼

X

 ref C in n ðK n;m þ I n;m Þ þ C n ðK n;m  I n;m Þ

ð5Þ ð6Þ

n

;1 ¼ A1 ðmÞC ref m

X tr C n ðK n;m  In;m ÞÞ

ð7Þ

n

where In;m ¼

Rd 0

  Rd  m n m n dy rnyz;tr um y;tr;1  uz;tr rzz;tr;1 , K n;m ¼ 0 dy ryz;tr;1 uy;tr 

n um z;tr;1 rzz;tr Þ can be considered as overlap integrals between the nth and mth modes of the adjacent (monolithic and threefold) regions, with AðnÞ; A1 ðmÞ normalization factors.

Y d

3. Mechanical displacement calculations in a plate with an inclusion

d2 Incident

Reflected d1

Transmitted

3.1. Parameters used for numerical implementation of the model 0

L

Fig. 1. Schematic view of Lamb wave incident on an inclusion in a plate.

Z

In this section, we implement the theoretical model for calculating mode conversion coefficients for the case of a non-absorptive steel plate that contains an inclusion with density, shear velocity

G. Shkerdin, C. Glorieux / Ultrasonics 53 (2013) 130–140

and longitudinal velocity similar to the respective values of steel. The density qst of steel is 7800 kg m3 , the shear velocity v s;st ¼ 3160 m s1 and the longitudinal velocity v l;st ¼ 5720 m s1 . Though we numerically consider particular choices of materials and acoustic parameters, the systematic analysis aims at gaining general insights on the dependence of the mode conversion coefficients on the characteristics of the inclusion and on the nature of the incident mode, and thus to understand better the interaction between Lamb modes and inclusions. From this, conclusions can be drawn concerning how to optimize the sensitivity of detecting inclusions by Lamb wave techniques by choosing a proper mode and frequency. Numerical calculations were performed for (virtual) kinds of (non-absorbing) inclusion material, which differ from steel in stiffness (or equivalenty, in acoustic wave velocity) as well as density. For the first inclusion material (‘type 1’) we used a material that is slower and lighter than steel, with qin ¼ 6000 kg m3 ; v s;in ¼ 3000 m s1 and v l;in ¼ 5000 m s1 : The second inclusion material (‘type 2’) material was taken to be even more slow and light, with qin ¼ 5000 kg m3 ; v s;in ¼ 2500 m s1 and v l;in ¼ 4500 m s1 : The third inclusion material (‘type 3’) was selected to be more dense and more slow than steel, with qin ¼ 10; 000 kg m3 ; v s;in ¼ 3000 m s1 ; v l;in ¼ 5000 m s1 and the fourth inclusion material (‘type 4’) was taken to be more light and fast, with qin ¼ 6000 kg m3 ; v s;in ¼ 3400 m s1 and v l;in ¼ 6400 m s1 . As mentioned in the introduction, the interaction of Lamb modes with an inclusion is easier to analyze for the case of an acoustically thin plate, when there is only a small number of propagating Lamb modes. For this reason we have chosen a combination of thickness d ¼ 1 mm and acoustic frequencies f 6 1 MHz for which the acoustic wavelength k is larger than d. In this kind of situation only two fundamental propagating Lamb modes exist, since the lowest Lamb mode cutoff frequency, fc;s ¼ v s;st =ð2dÞ ¼ 1:58 MHz, corresponding with standing shear waves in the normal direction between the two edges of the plate, is larger than 1 MHz, and thus larger than all considered frequencies. 3.2. Influence of the inclusion parameters on the A and S Lamb mode phase velocity It is well known that the amount of mode conversion due to an acoustical heterogeneity scales with the degree of acoustical impedance mismatch, which in turn scales with the degree of mismatch in density and wave velocity. Therefore, before tackling the issue of mode conversion of an incident mode at an inclusion, it is useful to take a look into the effect of the inclusion layer on the local wave mode velocities and profiles. Fig. 2a and b illustrate that due to the similarity in elastic properties, the phase velocities of the lowest propagating Lamb modes ðf ¼ 0:6 MHzÞ in the inclusion region, which was chosen to occupy 100 lm or 10% of the total plate thickness, are close to the ones of the fundamental S0 and A0 modes in the uniform region, for all inclusion layer positions throughout the plate. For this reason we denote these modes as respectively S and A modes, in spite of the fact that they are not perfectly symmetric or anti-symmetric in cases where the inclusion is asymmetrically located. Due to the symmetry of the structure under consideration the phase velocity curves are symmetrical around the plate (and coincident inclusion) center. In order to gain more insight in the dependence of the velocity changes on the inclusion location, we have also calculated the transverse profiles of the relative displacement ðuy and uz Þ and stress tensor ðryz ; ryy ; rzz Þ component amplitudes of the S and A Lamb modes, and of their power densities (I) in the threefold plate region (Fig. 3) for type 2 (dotted line) and type 4 (dash-dot

1

0

−1

Δ v/v (%)

132

−2 1 0 −1 −2 −3

0

0.2

0.4

0.6

0.8

1

Yinclusion center (mm) Fig. 2. Dependence of the relative effect (%) of an inclusion (type 1: dash-dot line, type 2: dotted line, type 3: full line, type 4: dashed line) on the S0-like (a) and A0like (b) Lamb mode velocity, on the position of the inclusion center across a 1 mm steel plate (with v S0 ¼ 5250 ms1 and v A0 ¼ 1986 ms1 for acoustic frequency f ¼ 0:6 MHzÞ. The considered inclusion layer has a thickness ðd2  d1 Þ of 100 lm, i.e. 10% of the total plate thickness.

line) inclusions of thickness equal to 100 lm ðd1 ¼ 0:7 mm and d2 ¼ 0:8 mm), and compared them with the corresponding profiles of the S0 (left) and A0 (right) fundamental Lamb modes in a homogeneous plate. For the sake of easy comparison, the mode amplitudes were taken such that all modes were carrying the same acoustical power. The depicted stress tensor components were normalized with the Lamé coefficient of steel, l ¼ qst v 2s;st . Compared with the S mode, the phase velocity of the A mode depends more strongly on the location of the inclusion position in the plate. This reflects the less uniform energy density distribution of the A mode, in particular towards the plate edges, due to its shorter wavelength. Comparing the effect of the inclusion position on the velocity with the spatial profile of the different wave quantities, whose geometrical symmetry is different between the A and S modes, also allows to get a grasp on which wave quantity is most determining the sensitivity to the presence of the inclusion. For the S-mode, the change of wave velocity due to the presence of the inclusion is monotonically increasing from the plate center towards both plate edges. A similar monotonic change away from the center is also observed for all wave quantities, except for the shear stress ryz , which is zero in the center as well as at the edges. For the A-mode, the change of wave velocity due the inclusion is first decreasing, and then strongly increasing again, as the inclusion is positioned further away from the plate center, and closer to the edges. This non-monotonic profile is also exhibited by ryy . From these two observations, one could conclude that for both S and A mode, the ryy profile is reflecting to a reasonable degree the sensitivity of the Lamb waves to the presence of the inclusion. Although this is valid for this particular geometry, more cases (in terms of spatial distribution of wave quantities) of guided wave– inclusion interaction would need to be examined in order to generalize these conclusions. It is also worth to note that, for the S mode, positioning the inclusion asymmetrically results in a drastic increase in magnitude and change in symmetry of the ryz ðzÞ profile,compared to the homogeneous case, in which the shear stress is very small compared to the normal stresses (Fig. 3). As expected, the velocity (Fig. 2a and b) is globally increasing when the inclusion is stiffer (i.e., the inclusion has larger material

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10

5

5

0

0

10

10

5

5

yz

|σ |

z

|u |

y

|u |

10

0

2

100

1

50 0

4

20

2

10

0

0

150

400

yy

|σ |

0

100

|I|

zz

|σ |

0

200

50 0

0

20

15 10 5 0

10 0

0

0.5

y (mm)

1

0

0.5

1

y (mm)

Fig. 3. Transverse profiles of the different Lamb mode quantities, juy j (a), juz j (b), jryz j (c), jryy j(d), jrzz j (e), I (f), for an S0 (left) and A0 (right) Lamb mode in a homogeneous 1 mm steel plate (full line), and in the plate containing an inclusion layer in the region ðd1 ¼ 0:7 mm; d2 ¼ 0:8 mm) made of type 2 (dotted line, slower and lighter than steel) and type 4 (dash-dot line, faster and lighter than steel) material. Although the vertical scales are in arbitrary units, graphs of quantities with equal dimensions have been scaled in the same way so that they can be inter-compared. Note that the vertically connecting lines in the graphs of jrzz j and jIj reflect a discontinuity at y ¼ 0:7 mm and y ¼ 0:8 mm. For the sake of clarity, the curves for the type 1 and type 3 materials, which lie in between the other ones, are not shown.

velocities and thus elastic moduli) than the plate (inclusion type 4), and decreasing when slower (inclusion types 1–3). Comparing the type 1 and 4 inclusions (with equal inclusion density), the effects on the wave velocity of higher (type 4) and lower (type 1) bulk velocities are not symmetric. Increasing (only) the density (compare type 3 to type 1) is slowing down the wave. For type 2 both the inclusion material velocities and the density are decreased compared to the surrounding steel, but apparently the guided wave velocity decreasing effect of the change of material velocity is dominating to the guided wave velocity increasing effect of the change of density. One can thus expect that in case of a corrosion inclusion, the guided wave velocity reduction due to an expected reduction of bulk velocities is enhanced by the expected reduction of the local density, making the corrosion defect easier to detect. 3.3. Numerical calculation and accuracy of calculated A and S Lamb mode conversion coefficients When a propagating Lamb mode enters the region containing the inclusion, then due to the elastic mismatch at the interfaces z ¼ 0; L, all possible propagating modes, as well as an infinite number of non-propagating local Lamb modes is excited, creating a complex and dynamic mechanical displacement field that can be expressed as an expansion in these eigenmodes (see, e.g., Eqs. (2) and (3)). In order to calculate the normal displacement amplitude uy at the plate surfaces y ¼ 0 and y ¼ d, it is necessary to first solve the system of linear Eqs. (4)–(7). This can be done numerically, by limiting the number of local modes to a finite amount. The resulting truncation error can be assessed by considering the mismatch d of the energy balance between the incoming wave on one hand, and the propagating reflected and transmitted modes on the other hand:

d ¼ ðPin  Ptr  P ref Þ=Pin

ð8Þ

where Pin is the incident acoustic energy flow, P tr and Pref are the transmitted and reflected acoustic energy flow, respectively. The equations to calculate these energy flows are given in Ref. [13]. The energy mismatch parameter d however does not reflect possible calculation errors of the non-propagating mode coefficients, since non-propagating modes do not carry energy in the non-absorbing structure under consideration. Therefore, it is useful to additionally assess the reliability of the calculation on the basis of quality of fulfillment of the boundary conditions at the inclusion interfaces. The mechanical displacement amplitudes must be continuous along the z coordinate at z ¼ 0; L for all y coordinates, so that the relative calculation error can e.g. be estimated by the following dimensionless parameters ce;f :

ce;f ¼ jDuy ðy ¼ e; z ¼ fÞ=uy;in ðy ¼ d; z ¼ 0Þj

ð9Þ

where f ¼ 0; L; 0 6 e 6 d; uy;in ðy ¼ d; z ¼ 0Þ is the normal mechanical displacement for the incident wave (normalization point is chosen to be y ¼ d and z ¼ 0Þ and Duy ðy ¼ e; z ¼ fÞ is the discontinuity mismatch of the normal displacement at the interfaces f ¼ 0; L between the inclusion region and the surrounding plate material. Although one could also consider the continuity fulfillment of the tangential displacement component uz or of the relevant stress components rzz ; ryy and rzy , here we have chosen only to evaluate the virtually experimentally observable quantities, i.e. the normal displacement component at the locations e ¼ 0 and e ¼ d at the top and bottom plate interfaces respectively. Typical dependences of the relative errors cd;0  c0 ; cd;L  cL and d on the number (N m Þ of modes taken into account are shown in Fig. 4 for different incident modes (either A0 (right) or S0 (left)) and different types of inclusions (type 1: top figures, type 2:

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G. Shkerdin, C. Glorieux / Ultrasonics 53 (2013) 130–140

bottom figures). Note that the calculation results for e ¼ 0ðc0;0 and c0;L Þ are similar to those for e ¼ d and not shown in these figures. The calculations were performed for acoustic frequency f ¼ 0:6 MHz, inclusion length L ¼ 150 lm; d1 ¼ 0:07 cm and d2 ¼ 0:08 cm. In all truncations, the lowest ranked local modes, sorted according to the modulus of their wave number, were taken into account. The number of modes taken into account was the same in all parts of the structure. It is important to mention that, in order to reach satisfactory convergence with increasing the number of modes in the expansion, local modes with both positive and negative real part of the wave number should be taken into account. In non-absorbing plate structures, local modes always exist in pairs with complex wave numbers qn ¼ a þ bi. For the inclusion and plate parameters of interest, i.e. for rather small inclusions (with respect to the acoustic wavelength and to the plate thickness) and for a rather small mismatch between the inclusion and plate parameters, about 40 modes were needed to reach the satisfactory displacement continuity level of ðc0 ; cL Þ 6 1%. For some very large signal contrast levels (higher than the ones discussed in this paper), the number of modes needed before convergence was reached got so high that convergence was preceded by the occurrence of numerical instabilities. The energy conservation error d is much smaller than the continuity error, which indicates the special role of the non-propagating modes, which are not taken into account in the energy balance, but which do matter very strongly for matching the continuity conditions. At first sight, as only propagating modes contribute to the energy balance, it could be expected that the convergence of the amplitudes of the propagating modes follows the convergence of the energy balance fulfillment, rather than fulfillment of the boundary conditions. In order to get a view on this, we have

also evaluated the dependences of the absolute values of the relative expansion coefficients for reflected and transmitted A0 and S0     ;tr in Lamb modes C ref on the number of modes taken into A0;S0 =C A0;S0  account, for different incident modes and different types of inclu       tr in  sions. The transmission factors C trA0 =C in A0  and C S0 =C S0  are close to unity for all types of inclusions under consideration. Therefore, in     Fig. 5 their variations are depicted via the quantities 1  C trA0 =C in A0      and 1  C trS0 =C in S0 . The convergence of the expansion coefficient goes along with the degree of mismatch (the error for type 2 material is larger than for type 1 material, due to the parameters of type 2 material being more different from steel than the ones of type 1 material). About 30–35 modes are needed to reach an accuracy of better than 1% on the expansion coefficients of the propagating modes. In absolute terms, the calculation error on the expansion coefficients is rather well represented by the error on the displacement continuity (and not by the too small energy conservation error), which consequently can be used in practice to decide on how many modes to take into account. The calculation errors were found to typically increase with increasing mismatch between the inclusion and plate material parameters (compare errors between type 1 and type 2 inclusion in Fig. 4). Type 3 and type 4 inclusions introduce smaller disturbances of acoustical fields in the inclusion regions compared to ones created by the second type of inclusions (Figs. 2 and 3), implying smaller calculation errors and better convergence of propagating modes amplitudes (for the sake of conciseness, these results are not shown here). In what follows, the numerical calculations of the normal mechanical displacement in and around the inclusion region have

6 4 2 0

0.6 0.4 0.2 0

−0.4

12

3.5 incident S0 type 2 inclusion

8

incident A0 type 2 inclusion

3

δ (%), γ0 (%), γL (%)

10

10δ (%), γ (%), γ (%) 0 L

1 0.8

−0.2

−2

6 4 2 0 −2

incident A0 type 1 inclusion

1.2

incident S0 type 1 inclusion

8

10δ (%), γ (%), γ (%) 0 L

100δ (%), γ0 (%), γL (%)

10

2.5 2 1.5 1 0.5 0

0

10

20

30

n

40

−0.5

0

10

20

30

40

n

Fig. 4. Dependence of the percentual calculation errors d (circles), c0 (triangles) and cL (asterisks) on the number of Lamb modes taken into account in the modal decomposition in a 1 mm steel plate with a 100 lm wide and 150 lm long inclusion ðd1 ¼ 0:07 cm and d2 ¼ 0:08 cmÞ of type 1 (top) and 2 (bottom) material, for an incident S0 (left) and A0 (right) mode. For an incident S0 and A0 mode encountering a type 1 inclusion, the d values have been multiplied by 100 and 10 respectively. For an incident S0 mode encountering a type 2 inclusion, the d values have been multiplied with 10.

135

tr

incident S0 type 1 inclusion

1

incident A0 type 1 inclusion

ref

0.5

−0.1

−0.5

4

tr

0.8

0

tr

0.6 0.4

incident S0 type 2 inclusion

0.2 0 −0.2

0

10

20

30

40

n

ref ref |,|α |, A0,A0 A0,S0

tr

1.5

1−|αA0,A0|, |αA0,S0|

ref

0

|α

ref ref |,|αS0,S0|, S0,A0

|α

0.1

tr

|αS0,A0|, 1−|αS0,S0|

ref ref |,|α |, S0,A0 S0,S0

2

tr

1−|α

0.2

|α

0.3

|α

|αA0,A0|,|αA0,S0|, 1−|αA0,A0|, |αA0,S0|

0.4

tr |, S0,A0

tr | S0,S0

G. Shkerdin, C. Glorieux / Ultrasonics 53 (2013) 130–140

3 2

incident A0 type 2 inclusion

1 0 −1

0

10

20

30

40

n

     tr   tr     ref      Fig. 5. Convergence of the calculated values for the rescaled (see text) expansion coefficients (%) aref S0:A0  (circles), aS0;S0  (triangles), aS0:A0 (asterisks),1  aS0:S0 (squares) for          ref   ref  tr tr     an incident S0 mode (left), and aA0:A0  (circles), aA0;S0  (triangles), 1- aA0:A0 (asterisks), aA0:S0 (squares) for an incident A0 mode (right)) with increasing number of Lamb modes (n) taken into account in the modal decomposition in a 1 mm steel plate with a 100 lm wide and 150 lm long inclusion ðd1 ¼ 0:07 cm and d2 ¼ 0:08 cmÞ of type 1 (top) and 2 (bottom) material.

been performed taking into account 43 right-going (and where applicable, 43 left-going waves) in every sub-region, corresponding with a set of 172 equations to be solved for the structure shown in Fig. 1.

different: the S0 mode is mainly polarized in the tangential direction, while the A0 mode induced motion is oriented mainly normal   to the surface: for the case under consideration, uS0 y ðy ¼ 0Þ=    A0  S0 A0 uz ðy ¼ 0Þj  0:15 and uy ðy ¼ 0Þ=uz ðy ¼ 0Þ  1:64. When quanti-

3.4. Inclusion induced displacement fringes

fying the effect of the inclusion by the relative change with respect     to the largest mode amplitude, i.e. by U y ðy ¼ 0; dÞ=uA0 y   1 for the     1 for the case of the incident A0 mode, and by U y ðy ¼ 0; dÞ=uS0 z case of the incident S0 mode, the impact of the defect is very similar between both cases. In practice though, the S0 mode is the preferred one to use for experimental detection of inclusions, since it allows to detect large the largest relative change in the experimentally observable normal displacement component. Although the inclusion region Fig. 6 is quite small ðL ¼ 150 lmÞ, the effect of the inclusion is quite obvious from inspecting the disturbed acoustic field in its neighborhood. The existence of inclusion-induced converted wave modes that interfere with the incoming mode ðz < 0Þ and with each other (everywhere) leads to clearly observable interference fringes in the normal displacement pattern, with a dominating fringe spacing that is clearly different between the region before and after the inclusion, thus allowing to localize the inclusion site. In the incoming region the presence of one incoming and two reflected wave modes leads to beating effects with three spatial periodicities: the standing wave pattern due to the right-going incoming mode and its left-going counter-propagating reflection (spatial period 2p=ð2qA0 Þ  0:165 cm for incident A0 mode and spatial period 2p=ð2qS0 Þ ¼ 0:437 cm for incident S0 mode), and additional fringes due to interference between these modes and the converted mode. Even in the case of an incident right-going S0 mode, due to the asymmetric positioning of the inclusion with respect to the middle of the plate, the reflected part

Using the approach described in Section 3.3, we can now calculate the degree of conversion of an incident Lamb mode entering an inclusion zone to other mode types, study the resulting mixed wave pattern, and look for links with all involved quantities of the uniform plate and of the inclusion region. It can be expected that the mismatch between the material velocities and density in both regions plays an important role, as well as the mismatch between the velocity of the incident Lamb wave in the uniform plate and the velocity of the guided wave mode in the inclusion region, which in turn depends on the width, length and position of the inclusion, as discussed in Section 3.2. Typical evolutions along the propagation direction of the normal mechanical displacement amplitude at the plate surfaces y ¼ 0; d are shown in Fig. 6 for an incident A0 mode and for an incident S0 mode in the vicinity of a type 1 inclusion located between d1 ¼ 0:07 cm and d1 ¼ 0:08 cm. The calculations were performed for an acoustic frequency f ¼ 0:6 MHz and inclusion length L ¼ 150 lm. As the total acoustic field is only slightly deviating from the one of the respective incident mode, only the relative change of the normal mechanical displacement amplitudes     U y ðy ¼ 0; dÞ=uA0;S0   1 with respect to the incident mode is plotted. y The found disturbance of the normal surface component of the acoustic field due to the inclusion is considerably larger for the incident S0 mode than for the A0 mode. It should be noted here that the polarization of the surface motion of these two modes are quite

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0.5 y=0

2

y=0

1 0

0

|uy/uy |−1 (%)

−2

y

A0

S0 y

|u /u |−1 (%)

−1

y=d

2

−0.5 0.5 y=d

1 0

0

−1 −2 −4

−2

0

2

z (mm)

−0.5 −4

−2

0

2

z (mm)

Fig. 6. Evolution along the propagation direction of the relative (%) normal mechanical displacement amplitude at the plate boundary y ¼ 0 (top) and y = d (bottom), for an incident S0 (left) and A0 (right) mode.

of the wave consists mainly of a left-going A0 contribution, so that the dominating spatial period roughly equals 2p=ðqS0 þ qA0 Þ  0:24 cm. In the region behind the inclusion, a fringe period 2p=ðqA0  qS0 Þ  0:53 cm results from the interference between (non-converted and converted) co-propagating, right-going A0 and S0 modes. In practical acoustic wave imaging or pulse-echo applications, the signal to noise ratio might be quite poor. In that case, the detection of the foreseen defect-induced changes of the order of a percent in the situation of interest can be quite challenging. However, in full-field imaging, a differential approach, comparing the wave field pattern with the one interpolated from the field pattern in the surrounding regions can enhance the defect contrast. In the case of a pulse-echo approach in which the defect is expected in the region around the ultrasound receiver, one can make use of bursts, so that even very small reflected wave packets can be discriminated and localized by applying temporal windowing. 3.5. Dependence of the mode conversion efficiency on the inclusion position and material parameters In order to get a more quantitative view on the effectiveness of different inclusions placed at different depth intervals ½d1 ; d2  to induce mode conversion in transmission and reflection for different incident wave modes, we define mode conversion coefficients based on both the respective expansion coefficients C tr;ref n;m and normal displacement amplitudes un;m y :





  tr;ref n m atr;ref uy ðy ¼ 0Þ=C in m uy ðy ¼ 0Þ m!n ¼ C n

ð10Þ

where the subscripts ðn; mÞ refer to the kind ðA0 or S0Þ of incident mode (subscript mÞ, or transmitted ðtrÞ or reflected ðref Þ mode (subscript nÞ respectively. Due to the symmetry of the structure the conversion coefficients are symmetrical around the center of the plate (see also symmetry of effect of inclusion on mode velocities in Fig. 2). The dependences of the largest percentual conversion coefficients atr;ref n;m on the type of incident mode ðA0; S0Þ, inclusion center position dcenter ¼ ðd1 þ d2 Þ=2, the inclusion thickness din , and the

inclusion material parameters (inclusion material types 1, 2, 4) are elucidated in Fig. 7, for an acoustic frequency of 0:6 MHz: The calculation results for the type 3 of inclusion are not shown because of a very large similarity with the type 4 results. 3.5.1. General tendencies Given the similarity between all types of inclusion material and the surrounding steel plate, for all cases, the acoustical mismatch introduced by the inclusion remains is small, so that the calculation errors are very small, and the main part of the energy remains within the same mode: atrA0!A0 and atrS0!S0 are close to unity. As expected, the mode conversion sublinearly increases with increasing cross-section (i.e. with the length and the width) of the inclusion (Fig. 8). A type 1 inclusion of only about 50–100 lm long (and 100 lm wide) leads to a normal displacement mode conversion degree of 1%, which should be possible to experimentally observe. As mentioned above (Section 3.4), the experimentally observable (normal displacement) fringe contrast induced by an inclusion is highest for an incident S0 mode, which is partially mode converted to reflected and transmitted A0 waves. This effect is easily observable due to the large normal displacement component of both A0 modes compared to the S0 mode. It is interesting that although this observable contrast between normal displacements can amount up to 20% (for inclusion of din ¼ 0:1 mm and ðL ¼ 0:5 mmÞ, the change in normal displacement amplitude is only about 2–3% of the maximum tangential displacement of the incident S0 mode. In terms of energy conversion, the inclusion effect is even less. With the total energy reflection coefficient r m for an incident mode mðm ¼ S0 or A0Þ defined as (see Ref. [10]):

 2     ref in 2 in  rm ¼ C ref S0 =C m  F S0 =F m þ C A0 =C m  F A0 =F m

ð11Þ

R       d m  , we have calculated that with F m ¼ Im 0 dy rm  rm um zz uz yz y for this case only 0.1% of the incident S0 mode energy is transferred into reflected energy. This explains why atrS0!S0 is very close to unity, but makes the large corresponding normal displacement contrast value of 20% quite peculiar and experimentally interesting.

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6 αref ,αref ,αtr A0,A0 S0,A0 S0,A0

αref (%) S0−>A0

40

20

0 20

4

2

αref ,αref ,αtr A0,A0 S0,A0 S0,A0

0

2

0

0.4

0.5

1

0.05

0.1 d (mm)

0.15

0.2

tr Fig. 8. Dependence of the mode conversion coefficients aref S0!A0 (dotted line), aS0!A0 (full line) and aref A0!A0 (dash-dot line) for A0 and S0 Lamb modes ðf ¼ 0:6 MHzÞ coming from a 1 mm uniform steel plate; (a) on the length L of an encountered region with a type 1 inclusion layer with thickness din ¼ 0:1 mm, and (b) on the width din of an encountered inclusion region with width L ¼ 0:1 mm. In both figures tr the inclusion center is located at dc ¼ 0:075 cm for the coefficientaref S0!A0 and aS0!A0 and at dc ¼ 0:095 cm for aref A0!A0 .

(%)

4

2

αref

A0−>A0

0.2 0.3 L (mm)

2

0 0

ref

αS0−>S0 (%)

4

0

0.1

3

10

αtr

S0−>A0

(%)

0 0

0

0.2

0.4

y

0.6

inclusion center

0.8

1

(mm)

Fig. 7. Dependence on the inclusion center position y across the plate, of the ref ref tr percentual conversion coefficients aref S0!A0 (a), aS0!A0 (b), aS0!S0 (c) and aA0!A0 (d) for f ¼ 0:6 MHz Lamb waves coming from a uniform 1 mm steel plate and entering a region with an inclusion of L ¼ 0:5 mm length and width din ¼ 0:1 mm, for a type 1 (dash-dot line), type 2 (dotted line) and type 4 (full line) inclusion.

3.5.2. Peculiar observations Interestingly, the conversion coefficient aref A0;A0 of an incident A0 mode into a reflected A0 mode strongly depends on the location of the inclusion (Fig. 7). The aref A0;A0 values for soft and light inclusions (type 1 and type 2) even become zero and change sign for a particular inclusion center depth dc;0 (between 0.082 and 0.085 cm), which shifts toward the plate center both for smaller inclusion length and for larger inclusion width. From this observation we can conclude that for soft and light inclusions the sign and magnitude of aref A0;A0 can serve as useful observables for estimating the depth of this inclusion type. The conversion coefficients aref A0;A0 for soft and heavy inclusions (type 3) and for stiff and light inclusions (type 4) strongly depend on the inclusion position as well, but the zero value is only reached at the plate edge. Given the mechanism behind mode conversion, the sign change for specific inclusion type should be explained by the particular distribution of the mechanical displacement and stress tensor components in the plate without and with inclusion. Since for an incident A0 mode the acoustic field in the region with the inclusion is dominated by the A mode, the reflected A0 mode is mainly determined by the term C tr;1 A0 ðI A0;A0  K A0;A0 Þ in Eq. (5). Similar to aref , the sign of this term is changed for soft and light (type 1 A0;A0 and type 2) inclusion types when the y-coordinate of inclusion center is about (0.082–0.085) cm, and it shifts towards the plate center both for smaller inclusion length and for larger inclusion thickness.

This particular observation allows to make the more general conclusion that the degree of mode-conversion by a heterogeneity from a given incident mode to a given converted mode is strongly connected with the magnitude of the difference between the two overlap integrals I and K, and thus with the mismatch between the modes on both sides of the interface between the modified region and its environment. Taking another point of view, the location at y ¼ 0:082—0:085 cm where due to soft and light (type 1 and type 2) inclusions aref A0;A0 changes sign also coincides with the maximum of the ryy ðyÞ profile of the A mode (Fig. 4d), and with the location where the dependence of the A mode velocity change on the inclusion position (Fig. 2b) reaches a an extreme value. The ryy ðyÞ profile and the mismatch between the guided wave mode velocity in the uniform and inclusion parts of the plate are thus useful quantities to consider when estimating the effect of an inclusion on mode conversion. 3.5.3. Inclusion material properties relevant for mode conversion efficiency The degree of mode conversion (Fig. 7) due to a type 2 inclusion (dotted line) is typically about 2 times larger than for a type 1 inclusion (dash-dot line). From this we can conclude that a larger mismatch in guided mode velocity between the inclusion region and the uniform plate (for the maximum effect of the inclusion at the edge in Fig. 2, the A mode velocity mismatch is 2.6% for a type 2 inclusion and 1.4% for a type 1 inclusion), combined with a larger mismatch in density (36% for type 2 inclusion and 23% for type 1 inclusion) leads to stronger mode conversion. From comparing the degree of mode conversion (Fig. 7) between the type 1 (inclusion material 5–13% slower than plate material, 1–12% mode conversion, dash-dotted line) and type 4 (inclusion material 8–12% faster than plate material, 0.4–4% mode conversion, full line) inclusion (which have the same density of 6000 kg m3) we can conclude that the magnitudes of the effects of stiffening and softening are not equal.

G. Shkerdin, C. Glorieux / Ultrasonics 53 (2013) 130–140

3

2

1

αref

A0,A0

,α

ref ,αtr S0,A0 S0,A0

(%)

138

0

0

0.2

0.4

y

0.6

inclusion center

0.8

1

(mm)

Fig. 9. Dependence of mode conversion coefficients (%) for different conversion  ref types aref S0!A0 andf ¼ 0:6 MHz: thin full line, aS0!A0 and f ¼ 1 MHz: thin dashed line,

atrS0!A0 and f ¼ 0:6 MHz: bold dashed line, atrS0!A0 and f ¼ 1 MHz: bold full line, ref aref A0!A0 and f ¼ 0:6 MHz: thin dotted line, aA0!A0 and f ¼ 1 MHz: thin dash-dot line) on the location across a 1 mm steel plate for a type 1 inclusion with din ¼ L ¼ 0:1 mm for different acoustic frequencies. The dependence of the different experimentally accessible quantities on the inclusion location is different, so that their relative magnitudes can be used to locate an inclusion across the plate.

Though not shown in the figures, we have also calculated the mode conversion due to a type 3 inclusion. Qualitatively the curves are close to the ones of the type 4 inclusion (Fig. 7). The type 3 inclusion has the same material velocities as the type 1 inclusion; both are slower than the steel plate (the differences between the A and S modes and the respective A0 and S0 modes in type 3 are on average about 1.5 times larger than in type 1 material), but contrary to type 1 material, type 3 is denser than type 1. For the same geometrical parameters, the mode conversion effect of the type 3 inclusion is 2–3 times smaller than the one due to type 1 material. From this we conclude that for type 3 the effect of a negative guided wave velocity mismatch (type 3 is slower than steel) can be (partially) compensated by a positive density mismatch (type 3 is denser than steel). For the type 1 inclusion the mode conversion due to the negative guided wave velocity mismatch is further increased by the negative density mismatch. In other words, the mode conversion efficiency scales with the mismatch of the product of the density times the guided mode velocity, which can be looked upon as a kind of effective acoustic impedance. 3.5.4. Inclusion characterization An important incentive for studying the interaction of Lamb modes with inclusions is the aim of experimentalists and researchers in non-destructive evaluation (NDE) methods to solve the respective inverse problem, i.e. to identify and determine the inclusion parameters (density qin , longitudinal velocity v l;in , transverse velocity v t;in , width din , length L, and center depth dcenter Þ from the experimentally probed surface displacement field. In order to achieve this goal, the possibility to choose the acoustic frequency can be exploited as follows. The change of sign of the aref A0;A0 mode conversion coefficient is clearly connected with the coincidence of the I–K overlap integral mismatch and the inclusion location, discussed in Section 3.5.2. By tuning the acoustic frequency, also the depth profile of the modes can be controlled (Fig. 9). In this way, once a theoretical or empirical calibration has been estabref A0;A0

lished, determining the magnitude and sign change of the a versus frequency allows to determine the cross-section and depth of the inclusion respectively. 4. Conclusions The conversion of an incident Lamb mode in a plate by a thin, rectangular inclusion, in perfect mechanical contact with the

surrounding plate material was calculated by means of a modal decomposition approach. The dependence of the mode conversion coefficients on the geometrical and material characteristics of the inclusion was determined. An asymmetric position of the inclusion typically leads to channeling of energy from symmetric into antisymmetric reflected and transmitted modes. For an incident antisymmetric A0 mode, the sign of the conversion coefficient to an antisymmetric A0 reflection depends on the depth of the inclusion, giving a possibility for depth localization. In general, the magnitude and sign of mode conversion coefficients is determined by the mismatch between the respective modes on both sides of the interface separating the inclusion region from its environment, as expressed by a difference between I and K overlap integrals. For an incoming A0 mode, the normal profile ryy ðyÞ turns out to play a crucial role in the dependence of the inclusion-induced mode conversion strength on the inclusion position y. Even an energetically moderate conversion from a mainly tangentially polarized incoming S0 mode to a mainly normally polarized A0 mode turns out to be effective to detect an inclusion, due to the large related normal displacement contrast between the A0 and S0 mode. The presence of mode converted waves can be easily observed due to fringes in the normal displacement pattern, which are resulting from the interference between the incident and converted modes. By comparing different types of inclusion materials, it was shown that the largest mode conversion efficiency scales with the mismatch between the inclusion region and the uniform plate region of an acoustic impedance like quantity, i.e. the product of the (inclusion/uniform plate) density times the (A/A0) Lamb wave velocity. Acknowledgments The authors thank the Research Council of the K.U. Leuven for the fellowship awarded to Professor G.N. Shkerdin, which enabled us to perform this research. The authors are also grateful for the support from EU project – Aircraft integrated structural assessment II (AISHAII) (EU-FP7 – CP21912) of the experimental research that has stimulated this work. Appendix A. Calculation of displacement and stress fields for Lamb modes in three layered structure For the calculation of the displacement fields of the displacement and stress fields in the incident ðz < 0Þ and the transmitted ðz > LÞ region, which can both be handled as ordinary plates, we refer to Refs. [14,15]. The displacement fields for the forward propagating (positive z-direction) Lamb mode in the first ð0 6 y 6 d1 Þ, second ðd1 6 y 6 d2 Þ and third ðd2 6 y 6 dÞ layer of the composite plate in the inclusion region 0 6 z 6 L can be written as follows:

  uy;1 ¼ Aþ1t qeiqty;1 y  A1t qeiqty;1 y þ Aþ1l qly;1 eiqly;1 y  A1l qly;1 eiqly;1 y eiðqzxtÞ ðA1Þ   uz;1 ¼ Aþ1t qty;1 eiqty;1 y  A1t qty;1 eiqty;1 y þ Aþ1l qeiqly;1 y þ A1l qeiqly;1 y eiðqzxtÞ ðA2Þ   uny;2 ¼ Aþ2t qeiqty;2 y  A2t qeiqty;2 y þ Aþ2l qly;2 eiqly;2 y  A2l qly;2 eiqly;2 y eiðqzxtÞ ðA3Þ   uz;2 ¼ Aþ2t qty;2 eiqty;2 y  A2t qty;2 eiqty;2 y þ Aþ2l qeiqly;2 y þ A2l qeiqly;2 y eiðqzxtÞ ðA4Þ   uy;3 ¼ Aþ3t qeiqty;3 y  A3t qeiqty;3 y þ Aþ3l qly;3 eiqly;3 y  A2l qly;3 eiqly;3 y eiðqzxtÞ ðA5Þ

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G. Shkerdin, C. Glorieux / Ultrasonics 53 (2013) 130–140

  uz;3 ¼ Aþ3t qty;3 eiqty;3 y  A3t qty;3 eiqty;3 y þ Aþ3l qeiqly;3 y þ A3l qeiqly;3 y eiðqzxtÞ ðA6Þ where q2ly;1 ¼ q2l;1  q2n , q2ly;3 ¼ q2l;3  q2n , q2ty;3 ¼

q2ty;1 ¼ q2t;2  q2n , q2ly;2 ¼ q2t;3  q2n , ql;1 ; ql;2 and

q2l;2



q2n ,

q2ty;2

¼

q2t;2

 q2n ,

ql;3 are the longitudinal

wave numbers in the first, second and third layers respectively, qt;1 ; qt;2 and qt;3 are the respective shear wave numbers, qn is the   wave number of the Lamb mode, A 1t;1l , A2t;2l and A3t;3l are the unknown coefficients to be found. Expressions for stress tensor components can be easily written down taking into account standard stress–strain relations (see, e.g. Ref. [15]). Taking into account stress-free boundary conditions at the free interfaces y ¼ 0; d as well as the boundary conditions at y ¼ d1;2 , where the mechanical displacement vector components uy;z and stress tensor components ryy;yz are assumed to be continuous due to perfect mechanical contact, a system of 12 linear equations for 12 unknown coefficients can be written down. The determinant of this homogeneous system of equations has to be equal to zero in order to allow non-zero solutions. This condition leads to the following dispersion equation for the Lamb modes wave number:

   g þ  g þ1 p2 =p1 g 2 þ g 1 p2 =p1  f f1þ  b1 2þ  b 2 2 þ  þ þ p2 þ p1 p2 =p1 p2 þ p1 p2 =p1    þ þ   g  g p =p g  þ g 1 p2 =p1 1 1 2 þ f2þ þ b2 2þ f1  b1 2þ ¼0 þ  þ    p2 þ p1 p2 =p1 p2 þ p1 p2 =p1

p1 ¼ a1 m1 þ qty;3 m3       m2 qty;3 þ m1 qn mþ3 m1 þ m3 mþ1 = mþ2 m1 þ m2 mþ1 ;

g 2 ¼ a1 m1 þ qn m4      þ m2 qn  m1 qly;3 mþ4 m1  m4 mþ1 = mþ2 m1 þ m2 mþ1 ;     g þ3 ¼ a2 a3l n1  qn n3 þ n2 qn  n1 qly;3 nþ3 n1 þ n3 nþ1 =  þ   n2 n1 þ n2 nþ1 Expðiqly;3 dÞ;     g 3 ¼ a2 a3l n1  qn n3 þ n2 qn þ n1 qly;3 nþ3 n1 þ n3 nþ1 =  þ   n2 n1 þ n2 nþ1 Expðiqly;3 dÞ;     g þ4 ¼ a2 n1 þ qn n4 þ n2 qn  n1 qly;3 nþ4 n1  n4 nþ1 =  þ   n2 n1 þ n2 nþ1 Expðiqly;3 dÞ;     g 4 ¼ a2 n1  qn n4  n2 qn þ n1 qly;3 nþ4 n1  n4 nþ1 =  þ   n2 n1 þ n2 nþ1 Expðiqly;3 dÞ; nþ1 ¼ qn b2 Expðiðqty;2  qly;2 Þd3 Þ þ qn c2 Expðiðqty;2 þ qly;2 Þd3 Þ þ qty;2 ;

ðA7Þ

þ þ           where f1 ¼ g  3  g 1 p3 =p1 ; f2 ¼ g 4  g 1 p4 =p1 ; b1 ¼ p3 þ p1 p3 =p1 ; þ   b2 ¼ pþ  p p =p , 1 4 1 4

pþ1 ¼ a1 m1 þ qty;3 m3       m2 qty;3  m1 qn mþ3 m1 þ m3 mþ1 = mþ2 m1 þ m2 mþ1

g þ2 ¼ a1 m1 þ qn m4      þ m2 qn þ m1 qly;3 mþ4 m1  m4 mþ1 = mþ2 m1 þ m2 mþ1 ;

n1 ¼ qn c2 Expðiðqty;2 þ qly;2 Þd3 Þ þ qn b2 Expðiðqly;2  qty;2 Þd3 Þ þ qty;2 ; nþ2 ¼ qly;2 b2 Expðiðqty;2  qly;2 Þd3 Þ þ qly;2 c2 Expðiðqty;2 þ qly;2 Þd3 Þ þ qn ; n2 ¼ qly;2 c2 Expðiðqty;2 þ qly;2 Þd3 Þ  qly;2 b2 Expðiðqly;2  qty;2 Þd3 Þ þ qn ; nþ3 ¼ a2l b2 Expðiðqty;2  qly;2 Þd3 Þ þ a2l c2 Expðiðqty;2 þ qly;2 Þd3 Þ  1;

pþ2 ¼ a1 a3t m1 þ qty;3 m4      þ m2 qty;3  m1 qn mþ4 m1  m4 mþ1 = mþ2 m1 þ m2 mþ1 ;

n3 ¼ a2l c2 Expðiðqty;2 þ qly;2 Þd3 Þ  a2l b2 Expðiðqly;2  qty;2 Þd3 Þ  1;

p2 ¼ a1 a3t m1  qty;3 m4       m2 qty;3 þ m1 qn mþ4 m1  m4 mþ1 = mþ2 m1 þ m2 mþ1 ;

n4 ¼ c2 Expðiðqty;2 þ qly;2 Þd3 Þ þ b2 Expðiðqly;2  qty;2 Þd3 Þ  a2t ;

pþ3





¼ 2 n1  qty;3 n3 þ n2 qty;3 þ  þ   n2 n1 þ n2 nþ1 Expðiqty;3 dÞ;

a

n1 qn

 þ   n3 n1 þ n3 nþ1 =

    p3 ¼ a2 n1 þ qty;3 n3  n2 qty;3  n1 qn nþ3 n1 þ n3 nþ1 =  þ   n2 n1 þ n2 nþ1 Expðiqty;3 dÞ;     pþ4 ¼ a2 a3t n1  qty;3 n4  n2 qty;3 þ n1 qn nþ4 n1  n4 nþ1 =  þ   n2 n1 þ n2 nþ1 Expðiqty;3 dÞ;

nþ4 ¼ b2 Expðiðqty;2  qly;2 Þd3 Þ þ c2 Expðiðqty;2 þ qly;2 Þd3 Þ  a2t ;

mþ1 ¼ qn b1 Expðiqly;1 d1 Þ þ qn c1 Expðiqly;1 d1 Þ þ qty;1 Expðiqty;1 d1 Þ; m1 ¼ qn c1 Expðiqly;1 d1 Þ þ qn b1 Expðiqly;1 d1 Þ þ qty;1 Expðiqty;1 d1 Þ; mþ2 ¼ qly;1 b1 Expðiqly;1 d1 Þ  qly;1 c1 Expðiqly;1 d1 Þ  qn Expðiqty;1 d1 Þ; m2 ¼ qly;1 c1 Expðiqly;1 d1 Þ þ qly;1 b1 Expðiqly;1 d1 Þ  qn Expðiqty;1 d1 Þ; mþ3 ¼ a1l b1 Expðiqly;1 d1 Þ  a1l c1 Expðiqly;1 d1 Þ þ Expðiqty;1 d1 Þ;

    p4 ¼ a2 a3t n1  qty;3 n4  n2 qty;3  n1 qn nþ4 n1  n4 nþ1 =  þ   n2 n1 þ n2 nþ1 Expðiqty;3 dÞ;

m3 ¼ a1l c1 Expðiqly;1 d1 Þ þ a1l b1 Expðiqly;1 d1 Þ þ Expðiqty;1 d1 Þ;

g þ1 ¼ a1 a3l m1  qn m3      þ m2 qn þ m1 qly;3 mþ3 m1 þ m3 mþ1 = mþ2 m1 þ m2 mþ1 ;

m4 ¼ c1 Expðiqly;1 d1 Þ þ b1 Expðiqly;1 d1 Þ  a1t Expðiqty;1 d1 Þ;

g 1 ¼ a1 a3l m1 þ qn m3       m2 qn  m1 qly;3 mþ3 m1 þ m3 mþ1 = mþ2 m1 þ m2 mþ1 ;

b2 ¼ ða2t a2l  1Þ=ð2a2l Þ;

mþ4 ¼ b1 Expðiqly;1 d1 Þ þ c1 Expðiqly;1 d1 Þ  a1t Expðiqty;1 d1 Þ;

¼ ða1t a1l  1Þ=ð2a1l Þ;

¼ ða2t a2l þ 1Þ=ð2a2l Þ;

c1 ¼ ða1t a1l þ 1Þ=ð2a1l Þ;

c2

b1

140

G. Shkerdin, C. Glorieux / Ultrasonics 53 (2013) 130–140

     a1 ¼ l3 q2t;3  2q2n = l1 q2t;1  2q2n ;      a2 ¼ l3 q2t;3  2q2n = l2 q2t;2  2q2n ;   a1t ¼ 2qn qty;1 = q2t;1  2q2n ;   a3t ¼ 2qn qty;3 = q2t;3  2q2n ;



      n n2 qty;3  qn n1 Expðiqty;3 dÞ = nþ2 n1 þ nþ1 n2 ; 

j ¼ cnþ1  qn ðExpðiqly;3 dÞ þ vExpðiqly;3 dÞÞ  qty;3 ðuExpðiqty;3 dÞ

  a2t ¼ 2qn qty;2 = q2t;2  2q2n ;

  nExpðiqty;3 dÞÞ =n1 ;

    a1l ¼ 2qn qly;1 = q2t;1  2q2n , a2l ¼ 2qn qly;2 = q2t;2  2q2n , a3l ¼ 2qn qly;3 =   q2t;3  2q2n ; l1 ; l2 and l3 are the Lamé coefficients of the first, the second and the third layer, respectively, and d3 ¼ d  d2 . All unknown coefficients can be connected to the coefficient Aþ 1t by the following expressions:   A1t ¼ Q = dm1 Aþ1t ðA8Þ    ðA9Þ Aþ1l ¼ b1  c1 Q = dm1 Aþ1t    þ  ðA10Þ A1l ¼ c1  b1 Q = dm1 A1t þ

þ

A2t ¼ A1t ðc=dÞExpðiqty;2 d2 Þ

ðA11Þ

A2t ¼ Aþ1t ð =dÞExpðiqty;2 d2 Þ þ þ A2l ¼ A1t ðb2 Expðiqty;2 d3  iqly;2 dÞ  c2  A2l ¼ Aþ1t ðc2 Expðiqty;2 d3 þ iqly;2 dÞ  b2

v

c c

þ

ðA12Þ

jExpðiqty;2 d3  iqly;2 dÞÞ=d ðA13Þ jExpðiqty;2 d3 þ iqly;2 dÞÞ=d ðA14Þ

þ

A3t ¼ A1t ðu=dÞExpðiqty;3 d1 Þ

ðA15Þ

A3t ¼ Aþ1t ðn=dÞExpðiqty;3 d1 Þ  þ  þ A3l ¼ A1t =d Expðiqly;3 d1 Þ A3l ¼ Aþ1t ð =dÞExpðiqly;3 d1 Þ

ðA16Þ ðA17Þ

v

ðA18Þ

where

 





  

v ¼ f2þ pþ2 þ pþ1 p2 =p1 þ b2 g þ2  g þ1 p2 =p1 = f2 pþ2 þ pþ1 p2 =p1





  b2 g 2 þ g 1 p2 =p1 ; 



 



u ¼  g þ2  g þ1 p2 =p1 þ v g 2 þ g 1 p2 =p1 = pþ2 þ pþ1 p2 =p1 ;   n ¼ g þ1 þ g 1 v þ pþ1 u Þ=p1 ; d¼

        m2 qn þ m1 qly;3 þ v m2 qn  m1 qly;3 þ u m2 qty;3  m1 qn  n m2 qty;3 þ m1 qn ; mþ2 m1 þ m2 mþ1









u n2 qty;3 þ qn n1 Expðiqty;3 dÞ



c ¼ n2 qn  qly;3 n1 Expðiqly;3 dÞ þ v n2 qn þ qly;3 n1 Expðiqly;3 dÞþ

Q ¼ dmþ1  qn ð1 þ vÞ  qty;3 ðu  nÞ: Substituting (A8)–(A18) into (A1)–(A6), the displacement fields for Lamb modes in the three layered structure are found. References [1] B.A. Auld, M. Tan, Symmetrical Lamb wave scattering at a symmetrical pair of thin slots, Ultrasonic Symp. Proc. (1977) 61–66. [2] S. Rokhlin, Diffraction of Lamb waves by a finite crack in an elastic layer, J. Acoust. Soc. Am. 67 (1980) 1157–1165. [3] S. Rokhlin, Resonant phenomena of Lamb waves scattering by a finite crack in a solid layer, J. Acoust. Soc. Am. 69 (1981) 922–928. [4] L. Wang, J. Shen, Scattering of elastic waves by a crack in an isotropic plate, Ultrasonics 35 (6) (1997) 451–457. [5] M. Castaings, E. Le Clezio, B. Hosten, Modal decomposition method for modeling the interaction of Lamb waves with cracks, J. Acoust. Soc. Am. 112 (2002) 2567–2582. [6] D.N. Alleyne, P. Cawley, The interaction of lamb waves with defects, IEEE Trans. UFFC 39 (3) (1992) 381–397. [7] M.J. Lowe, P. Cawley, J-Y. Kao, O. Diligent, The low frequency reflection characteristics of the fundamental antisymmetric Lamb wave a0 from a rectangular notch in a plate, J. Acoust. Soc. Am. 112 (2002) 2612–2622. [8] O. Diligent, M.J. Lowe, Reflection of the s0 Lamb mode from a flat bottom circular hole, J. Acoust. Soc. Am. 118 (2005) 2869–2879. [9] B. Hosten, L. Moreau, M. Castaings, Reflection and transmission coefficients for guided waves reflected by defects in viscoelastic material plates, J. Acoust. Soc. Am. 121 (2007) 3409–3417. [10] G. Shkerdin, C. Glorieux, Interaction of Lamb modes with delaminations in plates coated by highly absorbing materials, IEEE Trans. UFFC 54 (2) (2007) 368–377. [11] C. Glorieux, K. Van de Rostyne, K.A. Nelson, W. Gao, W. Lauriks, J. Thoen, On the character of acoustic waves at the interface between hard and soft solids and liquids, J. Acoust. Soc. Am. 110 (3) (2001) 1299–1306. [12] K. Van de Rostyne, C. Glorieux, W. Gao, W. Lauriks, J. Thoen, Experimental investigation of leaky Lamb modes by an optically induced grating, IEEE Trans. UFFC 49 (9) (2002) 1245–1253. [13] G. Shkerdin, C. Glorieux, Lamb mode conversion in an absorptive bi-layer with a delamination, J. Acoust. Soc. Am. 118 (4) (2005) 2253–2264. [14] G. Shkerdin, C. Glorieux, Lamb mode conversion in a plate with a delamination, J. Acoust. Soc. Am. 116 (4) (2004) 2089–2100. [15] B.A. Auld, Acoustic Fields and Waves in Solids, John Wiley & Sons, New York, 1973.