Effect of a functionally graded soft middle layer on Love waves propagating in layered piezoelectric systems

Ultrasonics xxx (2015) xxx–xxx

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Effect of a functionally graded soft middle layer on Love waves propagating in layered piezoelectric systems Issam Ben Salah a,⇑, Morched Ben Amor b,1, Mohamed Hédi Ben Ghozlen a a b

Laboratory of Physics of Materials, Faculty of Sciences of Sfax, BP 1171, 3000 University of Sfax, Tunisia Sfax Preparatory Engineering Institute, Menzel Chaker Road 0.5 km, BP 1172, 3000 Sfax, Tunisia

a r t i c l e

i n f o

Article history: Received 24 December 2014 Received in revised form 23 April 2015 Accepted 24 April 2015 Available online xxxx Keywords: Love waves Soft middle layer Functionally graded materials Stiffness matrix method ODE method

a b s t r a c t Numerical examples for wave propagation in a three-layer structure have been investigated for both electrically open and shorted cases. The first order differential equations are solved by both methods ODE and Stiffness matrix. The solutions are used to study the effects of thickness and gradient coefficient of soft middle layer on the phase velocity and on the electromechanical coupling factor. We demonstrate that the electromechanical coupling factor is substantially increased when the equivalent thickness is in the order of the wavelength. The effects of gradient coefficients are plotted for the first mode when electrical and mechanical gradient variations are applied separately and altogether. The obtained deviations in comparison with the ungraded homogenous film are plotted with respect to the dimensionless wavenumber. The impact related to the gradient coefficient of the soft middle layer, on the mechanical displacement and the Poynting vector, is carried out. The numericals results are illustrated by a set of appropriate curves related to various profiles. The obtained results set guidelines not only for the design of high-performance surface acoustic wave (SAW) devices, but also for the measurement of material properties in a functionally graded piezoelectric layered system using Love waves. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Since the invention of interdigital transducers (IDT) (such as filters, delay lines, oscillators, and amplifiers), several applications have emerged in the area of signal processing, and information storage [1,2]. Such devices were successfully utilized for transmitting and receiving surface acoustic waves SAW (such as Love wave). Love wave sensors are highly sensitive devices owing to the concentration of acoustic energy within a few wavelengths of the surface. For this purpose layered piezoelectric structures consisting of a piezoelectric top layer and an elastic substrate or an elastic layer coupled to a piezoelectric substrate have been adopted [3,4,6]. Additionally the manufacture of such high performance devices, layered structures involving functional materials are also considered. The propagation of Love wave in elastic or piezoelectric materials has been investigated by many researchers [3–6]. Li et al. [7] delivered his idea on the propagation of Love waves in functionally graded piezoelectric materials. Du et al. [8] investigated

⇑ Corresponding author. Fax: +216 74403934. 1

E-mail address: [email protected] (I. Ben Salah). Tel.: +216 74 241 403; fax: +216 74 246 347.

Propagation of Love waves in pre-stressed piezoelectric layered structures loaded with viscous liquid. Cao et al. [9] studied propagation of Love waves in a functionally graded piezoelectric material (FGPM) layered composite system. Furthermore, Liu et al. [10] investigated the Love waves in a smart functionally graded piezoelectric composite structure in which an FGPM layer is placed between a pure piezoelectric material layer and a metal substrate. Accordingly the propagation of Love waves in structures having bonding between top layer and substrate has become a research topic of great interest, with a particular focus on the effects related to the bonded interface as well as the utilized glue [11,12]. The effect of an imperfect interface on the SH wave propagating in a cylindrical piezoelectric sensor has been discussed by Li and Lee [13]. In the same context the effects of a soft middle layer on Love waves propagating in layered piezoelectric systems have been recently reported [14]. In practice and mainly in SAW devices, due to various causes such as micro-defects, damage, the glue applied to dissimilar materials may weaken the interfacial continuity and further affect the performance of the heterogeneous structure. An adequate description of such additional soft middle layer is useful and its effect cannot be neglected even if the middle layer is typically thin, usually several micro-meters thick.

http://dx.doi.org/10.1016/j.ultras.2015.04.011 0041-624X/Ó 2015 Elsevier B.V. All rights reserved.

Please cite this article in press as: I. Ben Salah et al., Effect of a functionally graded soft middle layer on Love waves propagating in layered piezoelectric systems, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.04.011

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The present work is motivated by recent contributions dealing with the effects of the elastic soft middle layer on wave propagation in layered piezoelectric structures by using a numerical approach [3,6,15–18]. Conversely to previous investigations mainly based on analytical methods [14,19], the numerical matrix method brings more flexibility, the gradient coefficient associated to electrical and mechanical properties of middle layer are assumed different. However when the propagating waves have more than one polarization, the analytical method seems unsuitable and no more solution can be extracted. In the present study, a three-layer structure model is proposed for investigating the effect of a soft middle layer on Love waves propagating in layered piezoelectric systems. The soft character implies that the shear wave velocity (VSH = 1041.51 m/s) of the middle layer is distinctly smaller than that of the upper piezoelectric layer (VSH = 1751.11 m/s). In Section 2 we describe a three-layer composite structure and the developed numerical methods (Stiffness matrix method (SMM), as well as the ODE approach) [15–18]. Since the soft middle layer physical properties are dependent on depth, the layer is stratified and a computationally stable recursive method for wave propagation is applied. An illustrative example is considered, it includes a piezoelectric layer (PZT-5H) deposited on an elastic substrate (SiO2) the same as [6]. This step is used to validate the developed codes under Matlab software. In Section 3 the description of dispersive behaviour is obtained under electrically open and short conditions. Additionally the effects of the middle layer thickness on phase velocity and electromechanical coupling factor are discussed. In the same way other aspects related to the Poynting vector and mechanical distributions are considered to bring more light on the system behaviour.

Fig. 1. A schematic configuration of a three-layer composite structure and coordinate system.

The normal stress vector ti3 = [t13, t23, t33] and the particle displacement u = [u1, u2, u3] are chosen as the six mechanical variables. For piezoelectric material the electric potential / and the normal electric displacement component D3, are chosen as two electric variables which must satisfy boundary relationships. This set of independent parameters is sufficient to describe the behaviour of any piezoelectric system. The eight-component state vector n = [u / ti3 D3]T for the piezoelectric material and soft middle layer, is utilized to write suitable constitutive equations of the whole system in the form of an ordinary differential equation system in x3 [16–18] :

@n ¼ i An @x3

ð2Þ

where A is the fundamental acoustic tensor [16–18]. 2. Theoretical background 2.3. Implementation for a graded soft middle layer

2.1. System description Consider wave propagation in a three-layer composite structure consisting of a transversely piezoelectric layer, a soft elastic middle layer and a half-space elastic substrate, as shown schematically in Fig. 1. For illustration a semi-infinite SiO2 substrate carrying a polythene layer in the middle and a piezoelectric layer PZT-5H on the top assumed hexagonal, has been adopted. The elastic and piezoelectric constants as well as the density and the dielectric constants, are reported in Table 1 [20]. Rectangular Cartesian coordinates (x1, x2, x3) are selected such that the x2-axis coincides with the polarization of Love wave. The Love mode becomes piezoactive with a high conversion rate, when the piezoelectric axis lies parallel to x2 axis. hf and hs are the thickness of the top piezoelectric layer and the soft elastic middle layer, respectively.

We here assume that all material properties of the soft middle layer have the same profile along the x3-axis direction with the following linear distribution: C ¼ C0 ð1 þ a x3 Þ, where a is the graded coefficient characterizing the degree of the material gradient in the x3 direction and the superscript ‘‘0’’ is attached to indicate the value of the property C in the neighbourhood of x3 = 0. C stands for any property among {Cijkl, eijk, eik, q}, these elements represent the elastic, piezoelectric, dielectric constants and the density, respectively. The superscript ‘‘0’’ denotes the property for the homogeneous material. But it must be reminded that the electrical and mechanical parameters may be assigned differently in magnitude. Though these material constants distributions are unrealistic, it would allow us to understand the influence of soft middle layer gradient upon the characteristics of wave propagation, and make use of it for designing more effective devices in practice.

2.2. Governing differential equations Let us consider a three layer composite structure (Fig. 1) and a harmonic plane wave propagating along the x1-axis of the form nðx3 Þ exp½iðk1 x1  xtÞ; where k1 is the projection of the wave vector along x1 the guiding direction. The approach in this paper consists to have a first-order equation, which permits to use linear systems concepts and the well-known properties of first-order ODE’s ‘‘Ordinary differential equation’’ and to satisfy interfacial boundary conditions in an extremely simple way [16–18]. The general solution for the state vector can be represented in this form as:

nðx1 ; x3 ; tÞ ¼ nðx3 Þ exp½iðk1 x1  xtÞ

ð1Þ

2.4. The boundary conditions To describe the Love waves in a three layer composite structure, the following boundary and continuous conditions should be satisfied. It should be pointed out that two kinds of electrical boundary conditions, electrically open and shorted conditions, would be taken into account in this study. As an example of electric boundary conditions, we consider either a shorted case (metalized) when potential is zero (/ = 0) or a free surface (non-metalized) when the charge density is zero (r = 0). The mechanical boundary condition for the external surface (x3 = h) is usually the stress-free condition (t23 = 0) with h = hs + hf.

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I. Ben Salah et al. / Ultrasonics xxx (2015) xxx–xxx Table 1 Material parameters used in calculations [20]. Materials

Elastic constants (109 N m2) C11

C33

C44

C12

C13

e15

e31

e33

e11

e33

q

PZT-5H Polythene SiO2

151 5.54 78.5

124 5.54 0.00

23 1.28 0.00

98 79.5 16.1

96 84.1 0.00

17 0.00 0.00

5.1 0.00 0.00

27 0.00 0.00

150 0.204 0.00

130.27 0.204 0.033

5700 1180 2200

Piezoelectric constants (C m2)

r refers to the charge density, it can be deduced from D3 (the normal electric displacement) values in the neighbourhood of the film free surface on both sides, [17,18]:

r ¼ D3 ðhþ Þ  D3 ðh Þ

ð3Þ

At the interface between the film and the middle layer at x3 = hs, both generalized stress and displacement vectors are assumed to be continuous. Also at the interface separating the middle layer from the substrate x3 = 0, both generalized stress and displacement vectors are assumed to be continuous. In the same at the interface separating the middle layer and the substrate, both generalized stress and displacement vectors are assumed to be continuous. 2.5. Stiffness matrix method for multilayered medium Let us consider a multilayered medium, consisting of a piezoelectric layer, a soft middle layer and a substrate (Fig. 1). The stiffness matrix method makes up a strong tool computationally stable to reduce the number of unknown amplitudes relatives to partial waves describing the dynamic state in one layer. This recursive method relates general stress and displacement vectors at the level of two successive interfaces [17,18]. Applying the recursive process from the bottom surface (x3 = 0) to the top surface (x3 = h), therefore the amplitudes in one layer are expressed in terms of the last layer which is fixed to the substrate. The total stiffness matrix KN which relates stress to displacement vectors at the interfaces x3 = h and x3 = 0 is obtained:



T h T0



¼ KN



U h U0





t i3

t i3 V

Mass density (kg m3)



r 

Dielectric constants (1010 F m1)

¼ M oc ½As ;

ð7Þ

¼ M sc ½As 

ð8Þ



For both cases Love velocity is the zero of Moc and Msc matrices. 3. Results and discussions The computational procedure is based both on the generalized stiffness matrix methods and the ordinary differential equation (Eqs. (7) and (8)). For the calculation of the phase velocity, the electromechanical coupling factor and the profiles of various magnitudes, we have used elastic and electrical properties of piezoelectric film and the elastic parameters, dielectric constants of soft middle layer and substrate (cf. Table 1) [20]. Since these characteristic of soft middle layer is function of x3, i.e. Cijkl = Cijkl(x3) and eij = eij(x3), the soft middle layer is divided into N layers with equal thickness and different characteristics. Each layer is considered as homogenous and the ODE method can be applied [16]. The calculation of surface wave velocity is performed according to the standard procedure [17,18], on the basis of SMM and continuity criterion. Even if the middle layer is non-piezoelectric, it is considered as piezoelectric with zeros for piezoelectric modulus, this is achieved to make homogeneous the electrical description of the whole system. That makes easier the application of electrical boundary conditions, and does not introduce any complication in the writing of the state vector.

ð4Þ 3.1. Comparison with published data

For making matrix developments easier KN is divided into four blocks: K N11 ; K N12 ; K N21 ; K N22 and the general stress and displacement vector at different interfaces, can be deduced in term of T0 and U0. Below the results are written according to As the set of amplitudes associated with the downward partial waves in the substrate. From boundary conditions eight dimensioned vectors T0 and U0 are replaced by Ts and Us respectively, which are four dimensioned vectors. The s subscript refers to the substrate [17].

T h ¼



  1  1 K N12  K N11  K N21  K N22  U s þ K N11  K N21  Ts

 1   U h ¼ K N21  T s  K N22  U s

ð5Þ

ð6Þ

Starting from the interface next to the bottom boundary, using recursively Eqs. (5) and (6), one obtains the state vector at all interfaces. Similarly for the state vectors inside a multilayered structure on a substrate, the stiffness matrix can be calculated once the state vectors at the top surface have been obtained. At this level, various physical amounts can be deduced according to {As} basis, mainly stress vector ti3, electrical potential / and charge density r, [17,18]. Either in the electrically shorted or open conditions, (4  4) characteristic matrix Msc and Moc can be constructed:

To check the validity of the formulations and programs, the approach was first implemented on a bi-layered plate. The structure is composed of a piezoelectric layer (PZT-5H) deposited on an elastic substrate (SiO2). The obtained numerical results agree very well with the analytical work given in Ref. [4] as shown in Fig. 2 for the fundamental mode. 3.2. Effect of homogeneous soft middle layer thickness on the phase velocity The dispersion curves for a three layer composite structure are shown in Fig. 3 simultaneously for both the electrically open ‘‘oc’’ and shorted cases ‘‘sc’’. The effect of thickness on the three first modes is shown in function of non-dimensional wave-number khf for selected thickness values of the middle layer hs. It is seen from Fig. 3 that for the fundamental mode, when the operating frequency increases the velocity of Love mode becomes close to the bulk wave of the SiO2 substrate. With decreasing the acoustic wave frequency, the velocity decreases until it converges to the velocity of the bulk wave of PZT-5H. Meanwhile, the phase velocity decreases as the thickness of the soft middle layer is increased. From this figure, it is noted that Love velocity is strongly

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I. Ben Salah et al. / Ultrasonics xxx (2015) xxx–xxx 3800

0.4 hs = 2 μ.m

our results Ref. [4]

3600

hs = 10 μ.m

0.3

3400

0.25 3200

K2

Phase Velocity (m.s -1)

hs = 5 μ.m

0.35

0.2

3000 0.15 2800 0.1 2600 0.05 2400

0

1

2

3

4

5

6

Non-dimensional wave number (kh f )

dependent on thickness of the soft middle layer. A similar phenomenon is revealed for the electrically shorted case ‘‘sc’’. In the low frequency range (khf < 3), the phase velocity undergoes minimum only for fundamental mode. For the second and third modes, the phase velocity exhibits comparatively high deviations between ‘‘oc’’ and ‘‘sc’’. In addition their corresponding cut-off frequencies are wide apart. 3.3. Effect of homogeneous soft middle layer thickness on the electromechanical coupling factor The other significant parameter is the coupled electromechanical factor K2 which plays an essential role in the design of SAW devices. For the ungraded combination system the deviation between Love velocities (Vph(oc)  Vph(sc)) rises with higher modes; accordingly an improvement of K2 is expected. This parameter is commonly defined for surface waves as [20]:

K 2 ¼ 2ðV phðocÞ  V phðscÞ Þ=V phðocÞ

ð9Þ

where, Vph(oc) and Vph(sc) are the phase velocities for the electrically open and shorted circuits, respectively. In Fig. 4 the variations of electromechanical coupling factor are plotted for some selected

Mode 3

h = 2 μ.m s h = 5 μ.m s h = 10 μ.m s

Mode 2

3400

0

0.5

1

1.5

2

2.5

3

3.5

4

Non-dimensional wave number khf

Fig. 2. Phase velocity dispersion curves for the layered piezoelectric deposited on elastic substrate; (solid line) the results from Du et al. [4] and (circle) the results of this paper.

3600

0

Fig. 4. Electromechanical coupling factor of the fundamental mode for selected values of thickness of homogeneous soft middle layer hs.

values of the homogeneous soft middle layer thickness hs. The obtained curve reveals that increasing the soft middle layer thickness increases the electro-mechanical coupling factor calculated for some selected values of hs, from 34.7% to 37.97%. The maximum value of the electromechanical coupling factor shifts to the low frequency area with an increase in the soft middle layer thickness. These results can be used to design different SAW sensors with high performance working at different frequency ranges by adjusting the extent of material properties. All the electromechanical tend to the same value when the frequency becomes sufficiently large. This means that the soft middle layer thickness has an insignificant influence on the energy propagation of the Love wave in the ultrahigh frequency zone. 3.4. Accuracy of the solution The numerical studies have been carried out based on the method of dividing the whole layer into a variable N sub-layers and treating each one as a homogenous layer. The stiffness matrix method, which has been proved to have good convergence and high stability, is employed for theoretical derivations. Fig. 5 presents the electromechanical coupling factor of the first mode, as calculated through the stiffness matrix method for different N (1, . . ., 11). The gradient coefficient has been fixed: a = 5. For this case, the convergence remains almost unchanged, while the error is so small that it can be neglected for N P 10. It can be seen that the curves of the electromechanical coupling factor obtained by the stiffness matrix method become very close when the number of sub-layers increases.

Phase velocity (m.s-1)

OC SC

3200

3.5. Effect of gradient coefficients of soft middle layer on phase velocity

Mode 1

3000

OC

2800

SC

2600

OC

2400 2200

SC

2000 0

1

2

3

4

5

6

7

8

Non-dimensional wave number (khf) Fig. 3. Phase velocity of Love waves plotted as a function of khf for selected values of thickness of homogeneous soft middle layer hs.

The dispersive curves in the functionally gradient structure resemble those of the homogeneous system. It shows that film stratification does not introduce a big change comparatively to homogeneous film results. That is why; the relative velocity change is obtained just after Love velocities are computed for homogeneous and stratified film. On the other hand, if the change in velocity is significant, the interaction between the surface waves and an electrode type transducer should be correspondingly greater. Therefore with the two sets of velocity data, the value of Dv/v is calculated in order to estimate the surface wave excitation efficiency by means of inter-digital electrode transducers. To highlight the influence of the gradient coefficient on phase velocity, the relative velocity of the first mode, DVoc/Voc, is plotted

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I. Ben Salah et al. / Ultrasonics xxx (2015) xxx–xxx 0.4

α M = -5 hs = 10 μ.m

3 x 10

N=1 N=9 N = 10 N = 11

-3

hs = 10 μ.m; α M = -3 hs = 10 μ.m; α M = -5

0.35

hs = 5 μ.m; α M = -3

2.5

hs = 5 μ.m; α M = -5 hs = 2 μ.m; α M = -3 hs = 2 μ.m; α M = -5

2

K2

Δ V oc / V oc

0.3

0.25

1.5

1 0.2 0.5

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0

2.4

0

Non-dimensional wave number khf

in Fig. 6 as a function of khf for selected values of gradient coefficient aM, while DVoc is the difference between the value of the phase velocity of Love waves in the functional graded and the homogeneous of soft middle layer. The reported plots reveal that the relative velocity change increases with increasing the mechanical gradient parameter. The obtained curves reveal a maximum for the relative velocity (equals 0.27%) shift appears at khf = 0.6. This maximum moves to the high frequency when the gradient factor becomes (aM = 3). The gradient coefficients of the soft middle layer affect both the electrical and mechanical properties. Prior these coefficients can be fixed separately, when electrical and mechanical gradient variations are applied separately. The reported plots reveal that the relative velocity change decreases with increasing the mechanical gradient parameter ‘‘aM’’. From Table 2, it is noted that the phase velocity depend on the soft middle layer thickness for the same parameter ‘‘aE’’. However, the gradient coefficient of the electrical parameter ‘‘aE’’ of the soft middle layer has a negligible impact on the phase velocity of Love waves propagating in the three-layer structures. 3.6. Effect of thickness on mechanical displacement fields We note that the displacement has been normalized in such a way that the displacement on the upper surface of the sensitive layer is equal to one. A variation of the normalized mechanical displacement (u2) with depth is shown in Fig. 7 for selected values of thickness of the soft middle layer (hs). This mechanical displacement distribution in a three layer homogeneous structure will be taken into account for fundamental mode and electrically open case. It is observed from Fig. 7 that the increase of the thickness of soft middle layer has a remarkable influence on the displacement, their amplitudes rise substantially in the film of PZT-5H and the value of the u2 within the soft middle layer decreases brusquely. From another point of view, the reported curves of mechanical displacement in Fig. 7 reveal a sharp attenuation when frequency becomes high; this behaviour is predictable for surface waves. Increasing the frequency involves confinement of energy near the free surface. 3.7. Effect of the gradient coefficients on mechanical displacement and Poynting vector The variations of the normalized mechanical displacement u2 versus the depth are shown in Fig. 8 for selected values of a. It is seen from Fig. 8 that the mechanical displacement in the substrate

1

1.5

2

2.5

3

3.5

4

Non-dimensional wave number khf Fig. 6. The relative velocity in electrically open case change DVoc/Voc plotted as a function non-dimensional wave number khf for selected values of thickness of hs and gradient coefficient affected in mechanical parameter aM.

Table 2 Fundamental mode phase velocity for selected values of middle layer thickness (hs) and varying values of electric parameter of middle layer for khf = 1.1.

aE = 5 aE = 3 aE = 0

Normalized mechanical displacement "u2"

Fig. 5. Electromechanical coupling factor for different number of sub-layers.

0.5

hs = 10 lm

hs = 5 lm

hs = 2 lm

2543.728 2543.752 2543.795

2620.062 2620.073 2620.094

2701.886 2701.890 2701.898

hs = 10 μ.m hs = 5 μ.m

1

hs = 2 μ.m 1 0.9

0.8

0.8 0.7 0.6

0.6

0.5 0.4

0.4

0.3 0.2 0.1

0.2

0

-5

-1

0

1

-4

-3

2

depth (m)

-2

-1

3

0

1

2

4

3

4

5 -4

x 10

Fig. 7. Normalized mechanical displacement u2 of the fundamental mode with depth for selected values of thickness of soft middle layer hs.

attenuates to zero within several wavelengths for the case a = 0 (i.e. ungraded middle layer). The material gradient has effect on the displacement distribution, in the middle layer and in the substrate, a decreases the penetration depth inside the piezoelectric layer. This means that material gradient coefficient can confine the transverse surface waves within the vicinity of the soft middle layer more efficiently which is in favour of designing SAW devices. On the other hand the Poynting vector is useful to show how the energy is flowing in the structure, for example the depth at which the greatest transmission of energy takes place, or the manner in which energy transmits from one layer to the adjacent one. Therefore, for a given mode, the power flow density vector at one point, also known as the Poynting vector, can be obtained by

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Normalized mechanical displacement (u 2 )

1

α = -5 α = -3 α =0

0.9 0.8

0.9

0.7

0.8 0.7

0.6

0.6

0.5 0.5

0.4

0.4

0.3

0.3

0.2

-2.5 -2 -1.5 -1 -0.5 0

0.5

1

1.5

2

2.5

0.1 0 -1

0

1

2

3

4

5

x 10

depth (m)

-4

Fig. 8. Normalized mechanical displacement u2 of the fundamental mode with depth for different values of gradient coefficient a, with hf = 100 lm and hs = 10 lm.

consisting of a piezoelectric layer, functionally graded middle layer and an elastic substrate. This work is successive to our previous work in which the Love wave propagation in the same structure was investigated, but the focus was placed on the effect of the soft middle layer thickness. Through the comparison between the effects of the material gradient and the middle layer thickness on the phase velocity and the electromechanical coupling factor of the Love wave propagation, we can conclude that the changes in phase velocity and electromechanical coupling factor induced by the change of thickness of the middle layer, which implies a potential factor for designing new surface wave devices with FGMs. The thickness and the mechanical gradient variation of the soft middle layer affect significantly the propagation of Love waves, whilst the electrical parameters have negligible influence. The thickness of soft middle layer has an important effect on both mechanical displacement and energy profiles in substrate, whereas, on the piezoelectric layer is insignificant. The results presented in this work are helpful for the design and application of surface acoustic wave devices. References

hs = 10 μ.m

1

hs = 5 μ.m hs = 2 μ.m

< π3 >

0.8

0.6

0.4

0.2

0 -1

0

1

2

depth (m)

3

4

5 -4

x 10

Fig. 9. Normalized power flow density vector of the fundamental mode versus depth for different values of hs, with hf = 100 lm.

time-averaging over a unit period. The component of interest is the 
 2 þ third one, it takes the following form: hp3 i ¼ Re T 23 @u @t
 @D3  u @t . The variations of hp3i versus the depth and for the first mode are shown in Fig. 9 for different thicknesses of the soft middle layer hs for electrically open. It is seen from Fig. 9 that the energy decays away from the upper interface of the soft middle layer (x3 = hs) abruptly. The increase of thickness of soft middle layer has no effect on the energy distribution inside the piezoelectric layer. While in the substrate it decreases in magnitude vs. the depth, which explains the peak value of electromechanical coupling factor K2 in the high thickness range of the soft middle layer. 4. Conclusions In this paper, we studied the effect of the soft middle layer thickness on the Love wave propagation in a three layer structure

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Please cite this article in press as: I. Ben Salah et al., Effect of a functionally graded soft middle layer on Love waves propagating in layered piezoelectric systems, Ultrasonics (2015), http://dx.doi.org/10.1016/j.ultras.2015.04.011