Dispersion characteristics of wave propagation in layered piezoelectric structures: Exact and simplified models

Wave Motion 96 (2020) 102559

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Dispersion characteristics of wave propagation in layered piezoelectric structures: Exact and simplified models ∗

Huangchao Yu a,b , , Xiaodong Wang b a b

Institute of Unmanned Systems, National University of Defense Technology, Changsha, Hunan, 410073, PR China Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8

article

info

Article history: Received 7 November 2019 Received in revised form 7 January 2020 Accepted 29 March 2020 Available online 3 April 2020 Keywords: Dispersion curve Piezoelectric Layered structure Wave propagation

a b s t r a c t This article presents a study of the dispersion characteristics of wave propagation in layered piezoelectric structures under plane strain and open-loop conditions. The exact dispersion relation is first determined based on an electro-elastodynamic analysis. The dispersion equation is complicated and can be solved only by numerical methods. Since the piezoelectric layer is very thin and can be modeled as an electro-elastic film, a simplified model of the piezoelectric layer reduces this complex problem to a non-trivial solution of a series of quadratic equations of wave numbers. The model is simple, yet captures the main phenomena of wave propagation. This model determines the dispersion curves of PZT4-Aluminum layered structures and identifies the two lowest modes of waves: the generalized longitudinal mode and the generalized Rayleigh mode. The model is validated by comparing with exact solutions, indicating that the results are accurate when the thickness of the layer is smaller or comparable to the typical wavelength. The effect of the piezoelectricity is examined, showing a significant influence on the generalized longitudinal wave but a very limited effect on the generalized Rayleigh wave. Typical examples are provided to illustrate the wave modes and the effects of layer thickness in the simplified model and the effects of the material combinations. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Structure health monitoring (SHM) plays a vital role in preventing catastrophic failures of critical structures, such as aircraft, pressure vessels and pipelines. Existing SHM systems are based mostly on the use of traditional transducers or their arrays [1,2]. These transducers cannot be organized in high density, and therefore provide only limited information for monitoring structures. Recently, significant attention has been paid to the development of new techniques to collect diagnostic elastic wave signals using piezoelectric materials and to realize continuous monitoring of structural integrity [3,4]. With the development and usage of highly sensitive piezoelectric sensors, a closely packed network of sensors can be bonded to structures. New developments in micro-fabrication technologies make it possible to add predesigned electrodes on the surface of thin-sheet piezoelectrics [5], which enables the collection of wave signals in a much higher density. The combination of the structure, the piezoelectric layer and the electrodes will form layered smart structures, for which the fundamental issue is how to evaluate their coupled electromechanical behavior and the dispersion characteristics of wave propagation. ∗ Corresponding author at: Institute of Unmanned Systems, National University of Defense Technology, Changsha, Hunan, 410073, PR China. E-mail address: [email protected] (H. Yu). https://doi.org/10.1016/j.wavemoti.2020.102559 0165-2125/© 2020 Elsevier B.V. All rights reserved.

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H. Yu and X. Wang / Wave Motion 96 (2020) 102559

The anti-plane problem of wave propagation (SH waves) in layered structures has been extensively studied. Typical works include the dispersion characteristics of Love waves in a piezoelectric lamina bonded to a semi-infinite solid medium [6] and the propagation of surface waves in piezoelectric coupled solids [7]. Although SH waves in such structures are well understood [8–12], due to the complexity of the problem the corresponding studies on in-plane wave propagation in layered piezoelectric structures are still relatively limited. Propagation of waves with long wavelengths in an elastic layer bonded to an elastic half space was first studied by Bromwich [13]. This work was then modified by Love in [14] by considering short waves, whose wavelengths were short compared to the thickness of the layer. Achenbach and Keshava studied the dispersion curves of free waves in an elastic plate supported by a semi-infinite elastic continuum [15]. Vinh et al. [16] derived the exact secular equations of Rayleigh waves in an orthotropic elastic half space overlaid by an orthotropic elastic layer. These works provided exact models for dispersion characteristics of layered elastic structures, but no piezoelectric materials are involved. Wave propagation in layered piezoelectric structures has also been studied, mainly focusing on Lamb waves and surface waves [17]. Lamb wave propagation in a dielectric semi-infinite medium with a piezoelectric layer has been studied using a simplified numerical solution of the dispersion curve by segmenting the phase velocity spectrum into different ranges [18]. The propagation behavior of surface waves in a layered pre-stressed piezoelectric structure with a thin piezoelectric layer bonded perfectly to an elastic substrate has also been numerically investigated [19], as have the dispersion characteristics of Rayleigh type surface waves in a transversely isotropic piezoelectric layer on a piezo-magnetic half-space [20] and the wave propagation in two layered piezoelectric plates [21]. These works provide useful information about the characteristics of elastic waves in layered structures, but this information derives mostly from complicated numerical solutions. For cases in which the layers are very thin, the approximate dispersion relations for waves in thin layers bonded to semi-infinite structures have been studied by modeling the layers as thin plates [15,22], or by expanding the displacements and stresses of the layers into Taylor series along the thickness of each layer [23,24]. Achenbach and Keshava obtained the approximate dispersion curves of free waves based on the Mindlin’s plate theory in [15] for an elastic layer. Vinh et al. developed an approximate secular equation of third order for the dispersion relation of Rayleigh waves [23], and then established a modified fourth order model [24]. These two models provided good approximate solutions, but can determine only the Rayleigh wave mode. It is, therefore, the objective of this study to determine a simplified analytical dispersion equation of wave propagation in layered structures with an elastic substrate and a surface-bonded piezoelectric layer as sensors. The exact dispersion equation will also be determined based on the elastodynamic analysis. Given that the thickness of such a piezoelectric layer is usually small (0.1–0.5 mm), a simplified theoretical model can be developed by modeling the piezoelectric layer as an electro-elastic film. This idea was first discussed, with some preliminary results, in one conference paper [25]. This paper documents more extensive studies that were conducted to further evaluate the two lowest modes of the guided waves. A closed form solution of the dispersion relation of this layered piezoelectric structure is also determined and compared to the results based on the simplified model, and the accuracy of the simplified analytical model is then analyzed. Two major wave modes are discussed in detail, and the influence of the layer thickness and the material property of the piezoelectric layer is studied. 2. Statement and formulation of the problem The problem examined in this study is the wave propagation in a layered piezoelectric structure, as shown in Fig. 1, which consists of a thin piezoelectric layer with uniform thickness h, as the sensor, and a homogeneous and isotropic elastic insulator as the substrate. It is assumed that the polarization direction of the piezoelectric layer is along the z-axis, perpendicular to the x–y plane. Since the thickness of the piezoelectric layer is usually very thin, much smaller than that of the substrate, the substrate can be treated as a half-space [26]. Harmonic in-plane waves of frequency ω are considered. In this case, the field variables, displacement, stress and strain, are all in the form of A(y, z , t) = A(y, z)e−iωt . −iωt

For convenience, the term e considered.

(1) will be omitted in the following discussion and only the magnitude A(y, z) will be

2.1. Governing equations The current piezoelectric structure with a bonded piezoelectric layer will, in general, undergo three-dimensional deformation when the out-of-plane dimension of the structure is finite. Considering that the out-of-plane dimension of commonly used piezoelectric thin sheets is significantly larger than their thickness, for the current problem it is assumed that the piezoelectric structure is under plane strain condition, and the upper and lower surfaces of the piezoelectric layer are short-circuited to generate an open-loop condition Dz = 0. In the following discussion, instead of solving for the ‘‘true’’ exact solution of the problem, these assumptions will be used to develop an exact solution for the plane strain problem under open-loop condition.

H. Yu and X. Wang / Wave Motion 96 (2020) 102559

3

Fig. 1. A thin piezoelectric layer surface-bonded to an elastic substrate.

The general governing equations of the piezoelectric layer are determined and given in Appendix A. Under the plane strain (y-z plane) and open-loop conditions, the governing equations can be reduced to 2 ) ∂ 2 uz ( ∗ ∂ 2 uy ∗ ∗ ∂ uy + c + c + c = −ρω2 uy 44 44 13 ∂ y2 ∂ z2 ∂ y∂ z 2 2 ( ∗ ) ∂ 2 uy ∗ ∂ uz ∗ ∗ ∂ uz + c33 + c13 + c44 = −ρω2 uz c44 2 2 ∂y ∂z ∂ y∂ z

∗ c11

(2)

with ∗ ∗ = c13 + e31 e33 /λ33 = c11 + e231 /λ33 , c13 c11 ∗ ∗ = c44 + e215 /λ11 = c33 + e233 /λ33 , c44 c33

where uy and uz are the displacements in y and z directions, respectively, cij are the stiffness parameters for a constant electric potential, eij are the piezoelectric constants, λij are the dielectric constants for zero strains, and ρ is the mass density of the piezoelectric layer. For the homogeneous and isotropic elastic substrate under plane strain condition, the governing equations can be obtained by setting the piezoelectric, dielectric constants in Eq. (2) to zero and c11 = c33 = λs + 2µs , c12 = c13 = λs , c44 = µs , and ρ = ρs , giving

) ( ) ( 2 ) ∂ 2 uy ∂ 2 uy ∂ 2 uy ∂ 2 uz 2 + c − c = −ω2 uy + + L T ∂ y2 ∂ z2 ∂ y2 ∂ y∂ z ( 2 ) ( ) ( 2 ) ∂ 2 uz ∂ uz ∂ 2 uz ∂ 2 uy 2 cT2 + c − c + + = −ω2 uz L T ∂ y2 ∂ z2 ∂ z2 ∂ y∂ z cT2

(

(3)

where cL and cT denote the √ √ velocities of longitudinal and transverse waves in the substrate, respectively, cL = (λs + 2µs )/ρs and cT = µs /ρs , in which µs , λs are the Lame’s elastic constants, and ρs is the mass density of the substrate. Eq. (3) is the same as the governing equation of elastic structure obtained in [15]. 2.2. Wave motion equations Consider free wave propagating in y direction in the piezoelectric layer, wich can be expressed in the form [27] uy = Ae−az eik(y−ct ) , uz = Be−az eik(y−ct ) ,

(4)

where k is the wave number, c is the phase velocity with c = ω/k, A and B are unknown constants, and a is a parameter to be determined. In the substrate, usy = As e−bz eik(y−ct ) , usz = Bs e−bz eik(y−ct ) , s

(5)

s

where A and B are unknown constants, b is a parameter to be determined. The real part of b is supposed to be positive, so that the displacements decrease with increasing z and tend to zero as z increases. Substituting Eq. (4) into Eq. (2) yields two homogeneous equations for constants A and B:

(

∗ ∗ ∗ 2 ∗ 2 c44 a − c11 k + ρ k2 c 2 A − i c13 + c44 akB = 0

)

(

)

(6)

( ∗ ) ( ∗ 2 ) ∗ ∗ 2 −i c13 + c44 akA + c33 a − c44 k + ρ k2 c 2 B = 0.

A nontrivial solution of this system of equations exists if and only if the determinant of the coefficients vanishes, which leads to the following eigen equation:

(

∗ 2 ∗ 2 c44 a − c11 k + ρ k2 c 2

)(

∗ 2 ∗ 2 ∗ ∗ c33 a − c44 k + ρ k2 c 2 + c13 + c44

)

(

)2

k2 a2 = 0.

(7)

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H. Yu and X. Wang / Wave Motion 96 (2020) 102559

The four roots of Eq. (7) are denoted as ai (i = 1, 2, 3, 4). The general solution of wave propagation in the piezoelectric layer can be expressed in the form uy =

4 ∑

Ai e−ai z eik(y−ct )

i=1

uz =

4 ∑

(8) −ai z ik(y−ct )

αi Ai e

e

i=1

where Ai (i = 1, 2, 3, 4) are unknown constants and αi (i = 1, 2, 3, 4) are,

αi = (B/A)i ,

(i = 1, 2, 3, 4) .

(9) s

s

In the substrate, by substituting Eq. (5) into Eq. (3), two homogeneous equations for A and B can be obtained as

[

cT2 b2 + k2 c 2 − cL2

(

)]

As − ikb cL2 − cT2 Bs = 0

(

)

(10)

( ) [ ( )] −ikb cL2 − cT2 As + cL2 b2 + k2 c 2 − cT2 Bs = 0.

A nontrivial solution of this system of equations exists if and only if the determinant of the coefficients vanishes, that is:

⏐ ⏐ c 2 b2 + k2 (c 2 − c 2 ) ⏐ T ( ) L ⏐ ⏐ −ikb cL2 − cT2

) ( −ikb cL2 − cT2 ( ) cL2 b2 + k2 c 2 − cT2

⏐ ⏐ ⏐ ⏐ = 0. ⏐

This equation can be reorganized as b2 − k2 (1 − c 2 /cL2 )

[

][

b2 − k2 (1 − c 2 /cT2 ) = 0

]

From which parameter b can be obtained as b1 = k(1 − c 2 /cL2 )1/2 , b2 = k(1 − c 2 /cT2 )1/2 .

(11)

Correspondingly, the general solution of the displacement of motion in the elastic substrate can be written in the form usy = (As1 e−b1 z + As2 e−b2 z )eiky ,

usz = (−

b1 ik

As1 e−b1 z +

ik b2

As2 e−b2 z )eiky .

(12)

2.3. Exact dispersion equations The exact dispersion equations will be obtained here for comparison purposes. The top surface of the piezoelectric layer is traction-free, and the displacements and stresses along the interface between the piezoelectric layer and the substrate are continuous; therefore, the boundary conditions are as follows: at z = −h

σzz = τyz = 0

(13)

at z = 0 s usy − uy = 0, usz − uz = 0, σzzs − σzz = 0, τyz − τyz = 0.

(14)

The general solutions given by Eqs. (8) and (12) are required to satisfy the boundary conditions, and the resulting equations are 4 ∑ (

∗ ∗ ikc13 − ai αi c33 Ai eai h = 0

)

(15)

i=1 4 ∑

∗ c44 (ikαi − ai ) Ai eai h = 0

(16)

i=1

As1 + As2 −

4 ∑

Ai = 0

(17)

i=1

−

b1 ik

As1 +

ik b2

As2 −

4 ∑ i=1

αi Ai = 0

(18)

H. Yu and X. Wang / Wave Motion 96 (2020) 102559

5

Fig. 2. The modeling of the piezoelectric layer.

(λs + 2µs ) [(

b21 ik

4

As1 − ikAs2 ) +

∑( ) λs ∗ ∗ Ai = 0 ik(As1 + As2 )] − − ai αi c33 ikc13 λs + 2µs

(19)

i=1

−µs (2As1 b1 + As2 b2 +

k2 b2

As2 ) −

4 ∑

∗ c44 (ikαi − ai ) Ai = 0

(20)

i=1

Eqs. (15)–(20) are a system of 6 homogeneous equations[ for] As1 , As2 and Ai (i = 1, 2, 3, 4). The dispersion equation can be obtained from the condition that the coefficient matrix Kjk (j, k = 1, 2, . . . , 6) of the system of equations is singular, or equivalently

⏐ ⏐ ⏐Kjk ⏐

6×6

=0

(21)

which is the exact dispersion equation of the problem; i.e. for a given value of wave number k, the phase velocity c can be solved through Eq. (21). It should be noted that this exact dispersion equation is based on the assumption that Dz = 0 in the layer under plane strain condition. This exact dispersion equation is complicated and most likely can only be solved by numerical methods. Considering that the piezoelectric layer is very thin, simplified analytical models can be developed to determine the lowest modes of wave propagation in this layered structure. 3. Simplified analytical model 3.1. Modeling of the piezoelectric layer For a thin piezoelectric layer bonded to an elastic half-space, its axial stiffness along the layer will play a more important role than its flexural stiffness. As a result, the piezoelectric layer can be modeled as a thin film with no bending stiffness [28]. The interfacial shear stress (τ ) transferred between the piezoelectric layer and the substrate can be treated as a distributed body force along the layer [29]. Therefore, the layer can be modeled as an electro-elastic thin film subjected to a distributed axial force τ , equivalent to a body force τ /h, and normal force σz , as shown in Fig. 2. The equation of motion of the layer can be expressed as dσy dy

+

τ (y) h

+ ρω2 uy = 0

(22)

σ z + ρ h ω 2 uz = 0

(23)

where σy is the axial stress, uy and uz are the axial and transverse displacements, and ρ is the mass density. The interface stress σz acts as the transverse load from the host, while the normal stress in z direction remains, to be approximately zero inside the layer. This is a commonly used model for beam-like structures. For example, in the vibration of a cantilever beam subjected to distributed pressure, the beam has no transverse normal stress but can carry the transverse load. For the current problem, when the bending stiffness of the layer is ignored, the equation of the transverse motion of the layer becomes Eqs. (22) and (23). In this case, the normal stress transferred between the layer and the substrate is equal to the inertia of the piezoelectric layer in z-direction. The constitutive relation of the piezoelectric layer under plane strain and open-loop conditions can be described, as shown in Appendix B, as

σy = Eeff

∂ uy , ∂y c2

Eeff = E +

e2

(24)

λ c

where E = c11 − c13 , e = e13 − e33 c13 , λ = λ33 + 33 33 and λij are the dielectric constants.

e233 c33

. cij are the stiffness parameters, eij are the piezoelectric constants,

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H. Yu and X. Wang / Wave Motion 96 (2020) 102559

The piezoelectric layer can be treated as a waveguide. For a free wave that propagates along the positive y-axis with a velocity c, the displacement can be described as uy = uy eiky , k > 0

(25)

where k = ω/c is the wave number and uy is the amplitude of the displacement. Substituting (24) and (25) into (22), the interfacial shear stress can then be related to the displacement uy as

τ = (k2 − where cs =

ω2 cs2

)Eeff huy

(26)

Eeff /ρ . The interfacial normal stress can be determined as

√

σz = −ρ hω2 uz

(27)

3.2. The simplified dispersion equations The general solution of displacements of a surface wave of velocity c in the substrate is given by Eq. (12). The shear and normal stress at the interface between the piezoelectric layer and the substrate can then be determined as

⏐ k2 τyzs ⏐z =0 = −µs (2As1 b1 + As2 b2 + As2 )eiky

(28)

⏐ b2 σzs ⏐z =0 = E ∗ [( 1 As1 − ikAs2 ) + ν ∗ ik(As1 + As2 )]eiky

(29)

b2

ik

with E ∗ = λs + 2µs , ν ∗ = λs /(λs + 2µs ). The continuity of displacements and stresses along the interfacez = 0can be expressed as b1

uy = (As1 + As2 )eiky , uz = (−

⏐ τyzs ⏐z =0 = τ ,

ik

ik

As1 +

b2

As2 )eiky

(30)

⏐ σzs ⏐z =0 = σz .

(31)

By substituting Eqs. (26)- (30) into Eq. (31), the boundary conditions can be expressed as

[(k2 −

[

b21 k

ω2 cs2

)Eeff h + 2µs b1 ]As1 + [(k2 −

− ν∗k −

ρ hω 2 b 1 E∗

(

k

ω2 cs2

)Eeff h + µs (b2 +

)]As1 + [k(1 − ν ∗ ) −

ρ hω 2 k E∗

(

b2

k2 b2

)]As2 = 0

)]As2 = 0.

(32)

(33)

The two equations for As1 and As2 have a non-zero solution when and only when the determinant of the coefficient matrix vanishes, which leads to the following dispersion equation:

⏐ ⏐ 2 ω2 ⏐ (k − )Eeff h + 2µs b1 ⏐ cs2 ⏐ ⏐ 2 ρ hω2 b1 ⏐ b1 ⏐ − ν∗k − ( ) ∗ k

E

k

(k2 −

ω2

2 )Eeff h + µs (b2 +

cs

k(1 − ν ) − ∗

ρ hω2

( bk ) E∗ 2

By substituting ω = kc , b1 = k(1 − c 2 /cL2 )1/2 and equation can be written in a polynomial of kh as

[

Eeff ρ c 2

(1 − c /

][

(1 − c /

k2 ) b2

1

⏐ ⏐ ⏐ ⏐ ⏐ = 0. ⏐ ⏐ ⏐

b2 = k(1 − c 2 /cT2 )1/2 into the above equation, the dispersion

]

(kh)2 + ⎧ ⎫ 2µs ρ c 2 (1 − c 2 /cL2 )1/2 Eeff (1 − ν ∗ ) (1 − c 2 /cs2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ∗ 2 + ⎪ ⎪ 2 2 /c 2 )1/2 ⎪ ⎪ E h h (1 − c ⎪ ⎪ T ⎪ ⎪ [ ] ⎨ ⎬ 2 2 2 µs ρ c (1 − c /cL ) 1 2 2 1/2 (kh) + + (1 − c / c ) + T ⎪ E ∗ h2 ⎪ ⎪ (1 − c 2 /cT2 )1/2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ) [ ]( ⎪ ⎪ ⎪ ⎪ ⎩ + Eeff (1 − c 2 /cL2 ) − ν ∗ 1 − c 2 /cs2 ⎭ 2 h { [ ][ ]} (1 − c 2 /cL2 ) − ν ∗ µs 1 − ν∗ 1 2 (1 − c 2 /cL2 )1/2 − (1 − c 2 /cT2 )1/2 + =0 h h h (1 − c 2 /cT2 )1/2 E ∗ h2

2

cs2 )

2

cL2 )

+

(34)

(1 − c 2 /cT2 )1/2

(35)

H. Yu and X. Wang / Wave Motion 96 (2020) 102559

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Table 1 Material constants of the piezoelectric layer. Elastic stiffness parameters ×1010 Pa Piezoelectric constants C/m2 Dielectric constants ×10−9 C/Vm Density ρ 7.5 × 103 Kg/m3

c11 13.2 e31 −4.1

c12 7.1 e33 14.1

λ11

λ33

7.1 Thickness h 0.001m

5.8

c13 7.3 e15 10.5

c33 11.5

c44 2.6

Table 2 Material constants of the substrate. Substrate

Young’s modulus E

Poisson ratio v

Density ρs

Constants

6.91 × 1010 N/m2

0.33

2700 Kg/m3

If piezoelectric layers disappear (h=0), the problem is degenerated to wave propagation along the free surface of an elastic half-space, which is a Rayleigh wave. In this case, Eq. (34) can be simplified to

⏐ ⏐ ⏐ 2µs b1 ⏐ ⏐ ⏐ b2 ⏐ 1 − ν∗k ⏐ k

µs (b2 +

k2 b2

k(1 − ν ) ∗

⏐ ⏐

) ⏐⏐

⏐=0 ⏐ ⏐ ⏐

(36)

which can be expressed as 2(1 −

c2 cL2

)1/2 (1 −

c2 cT2

)1/2 − (2 −

c2 cT2

)(1 −

c2 2ν cL2 11−−ν

) = 0.

(37)

Substituting the ratio of the transverse and longitudinal velocities, cL2 /cT2 = 2(1 − ν )/(1 − 2ν ), into Eq. (37), the dispersion equation can be simplified as 4(1 −

c2

c2

c2

cL

cT

cT2

)1/2 (1 − 2

)1/2 − (2 − 2

)2 = 0

(38)

which is exactly the dispersion equation of the Rayleigh wave [27]. 4. Comparison and results The dispersion equation describes the relation of the velocity of a wave to its frequency or wave number. The exact and simplified dispersion equations have been determined. For a specific phase velocity, the exact dispersion equation is a nonlinear equation in kh, while the simplified dispersion equation is quadratic in kh. The exact dispersion equation can be solved using typical numerical methods such as Newton’s method, and the simplified equation can be solved explicitly for kh in terms of c. This study considers only solutions corresponding to the propagating wave modes. 4.1. Dispersion curves of exact and simplified models The dispersion curves of propagating waves can be determined by solving the dispersion equations obtained. This section studies wave propagation in structures with a PZT4 layer and aluminum substrate. The material properties are given in Tables 1 and 2, while the resulting dispersion curves from both simplified and exact models are shown in Fig. 3. The simplified model can predict the two lowest modes of wave propagation in these layered structures, which are the most important modes of the wave. The lower branch mode starts from a velocity of 2884m/s when the wave number is 0. The Rayleigh wave velocity for the substrate, the aluminum half space, can be determined from Eq. (38) to be 2889 m/s. The lower branch is a generalized Rayleigh mode, which starts from a velocity close to the Rayleigh wave of the substrate at k = 0 and decreases with the increase of wave number. On the other hand, the phase velocity of the upper mode also decreases with the increase of wave number. The comparison shows very good agreement when the thickness of the layer is small compared to the wavelength (kh < 1), while the error between the current results and the exact solutions will become greater with the increase of kh. As expected, this model is suitable for cases in which a thin layer is bonded to an elastic substrate, which is consistent with the assumption of the current model. The exact model assumed Dz = 0. Considering that the electric field has limited effect on the mechanical field, this exact model will be very close to the true exact solution. The simplified model contains more assumptions, such as modeling the piezoelectric layer as a thin film with no bending stiffness, so it cannot

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H. Yu and X. Wang / Wave Motion 96 (2020) 102559

Fig. 3. Dispersion curves for the PZT4-Aluminum layered structure: exact and simplified solution.

accurately describe the high order stress distribution across the thickness, as with the limitation in general beam models. Therefore, the results from the simplified model are suitable only for relatively low frequencies (low kh), and can only describe low order modes, as the results demonstrate. The typical thickness of commonly used piezoelectric thin-sheets is in the range of 0.1 mm to 0.5 mm [28]. Therefore, in order to guarantee the accuracy of the result (kh < 1), the corresponding shortest wavelength the current simplified model can handle (when kh = 1) is about 0.6 mm to 3.1 mm if these thin sheets are used, which will provide reasonable detection resolution for structural health monitoring. 4.2. Physical explanation of the two modes In order to understand the physical meaning of the two branches of the dispersion relation, the effects of the mass density and the transverse inertia of the piezoelectric layer will be studied in detail. Based on the current analytical model, the dispersion curves of PZT4-Aluminum layered structures with different piezoelectric layer densities can be obtained, as demonstrated in Fig. 4. The density of the piezoelectric layer has a significant influence on the dispersion curves; i.e., the phase velocity will increase with the decrease of the density of the piezoelectric layer. When the density approaches zero, the lower branch approaches that of the Rayleigh wave. Therefore, the wave of the lower branch represents a generalized Rayleigh wave. When the piezoelectric layer is ignored, it will degenerate to a Rayleigh wave, as discussed earlier. The upper branch disappears when the density approaches zero. Fig. 5 shows√that the upper branch of dispersion curves will approach the effective longitudinal velocity of the piezoelectric layer cs = Eeff /ρ when kh → ∞. This indicates that the upper branch apparently degenerates to longitudinal waves in the layer along the y-axis when kh → ∞. In order to further evaluate the upper branch, the transverse inertia of the piezoelectric layer is ignored in the dispersion equation. In this case, the only non-zero stress component in the layer is σy , with σzs = σz = 0, which results in b21

+ ν ∗ ik)A1 − ik(1 − ν ∗ )A2 = 0. (39) ik Eqs. (32) and (39) can be reorganized in a matrix form from which the dispersion equation can be determined and simplified to the following dispersion equation: (

(2µs α k + Es hγ )(1 − ν ∗ ) + [µs k(β +

1

β

) + Es hγ ](ν ∗ − α 2 ) = 0

where α = (1 − c 2 /cL2 )1/2 , β = (1 − c 2 /cT2 )1/2 , γ = k2 −

(40)

ω2 cs2

As shown in Fig. 6, removing transverse inertia did not change the upper branch, indicating that the upper branch represents only a longitudinal wave mode along the y-direction, while the lower branch represents a Rayleigh-like mode. In order to show the generalized Rayleigh wave in the substrate, the displacement field is illustrated for a specific point at the dispersion curve of our solution, c = 2000 m/s, k = 1040/m. In this case, when the magnitude of usy is assumed to be unity, the real part of the normalized displacement field in the substrate at t = 0 can be expressed as real(usy ) = 26.3(e−984.4z − 0.962e−794.6z ) cos (1040y)

(41)

H. Yu and X. Wang / Wave Motion 96 (2020) 102559

9

Fig. 4. Dispersion curves for different densities of the piezoelectric layer (upper curves: generalized longitudinal waves; lower curves: generalized Rayleigh waves).

Fig. 5. Dispersion curves of the upper branch approach the effective longitudinal velocity of the layer cs .

Fig. 6. Dispersion curve when the inertia of the piezo-layer in z-direction is ignored (upper curves: generalized longitudinal waves; lower curves: generalized Rayleigh waves).

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H. Yu and X. Wang / Wave Motion 96 (2020) 102559

Fig. 7. The displacement wave field in y-direction when k = 1040/m, c = 2000 m/s.

Fig. 8. The displacement wave field in z-direction when k = 1040/m, c = 2000 m/s.

real(usz ) = −26.3(0.9465e−984.4z − 1.2591e−794.6z ) sin (1040y)

(42)

The normalized displacement fields in the substrate are plotted in Figs. 7 and 8. At the interface, the displacements are the same as those in the piezoelectric layer. In the substrate, the wave shows clearly similar properties to those of a Rayleigh wave, with the amplitudes decaying exponentially with the increase of depth. To summarize, the current simplified model can determine the two lowest modes of wave propagation in the layered piezoelectric structure. The first mode is a generalized Rayleigh wave propagating in the layered structure, while the second mode is the generalized longitudinal wave, which is mainly propagating along the layer and affected by the supporting substrate. 5. Discussion This simplified model can reasonably predict the two lowest wave modes in the layered structure. Because it is analytical, the model is useful and convenient for analysis of complicated wave phenomena, such as the effects of piezoelectricity and the material combinations. 5.1. Effect of piezoelectricity Dispersion curves for the PZT4-Aluminum structure with and without the piezoelectric effect are plotted in Fig. 9. The piezoelectric effect shows a significant influence on the generalized longitudinal wave, but an insignificant effect on the generalized Rayleigh wave propagation.

H. Yu and X. Wang / Wave Motion 96 (2020) 102559

11

Fig. 9. Comparison of dispersion curves with and without the piezoelectric effect (upper curves: generalized longitudinal waves; lower curves: generalized Rayleigh waves).

Fig. 10. Effect of the stiffness ratio on the dispersion curve (upper curves: generalized longitudinal waves; lower curves: generalized Rayleigh waves).

5.2. Effect of material combinations The dispersion curves for different mass densities are studied and given in Fig. 4, from which the physical meaning of each mode is clearly demonstrated. According to a previous analysis in [15], the dispersion curves will be affected by the stiffness ratio, which is defined as the ratio of Young’s modulus of the substrate (Es ) to the effective Young’s modulus of the piezoelectric layer (Eeff ). When the material properties of the layer are given and fixed, the dispersion curves for different Young’s moduli of the substrate can be obtained, as seen in Fig. 10. The generalized Rayleigh mode wave is only affected by the stiffness ratio to a limited extent. For the generalized longitudinal mode wave, the phase velocity will increase significantly with the decrease of the stiffness ratio. When the substrate is much softer than the layer, only one mode will exist, and the upper mode will disappear. The √ critical value for the existence of the upper mode is determined √ by the ratio of the longitudinal velocity of the layer, cs = Eeff /ρ , to that of the substrate, cL = (λs + 2µs )/ρs . As shown in Fig. 11, the upper mode exists when cL /cs > 1, but the upper mode disappears and only the lower mode exists when cs > cL . 6. Conclusions This paper developed a simplified model of how waves propagate in a semi-infinite elastic substrate with a surfacebonded piezoelectric layer. The current model is based on the assumption that the piezoelectric layer can be modeled as

12

H. Yu and X. Wang / Wave Motion 96 (2020) 102559

Fig. 11. Dispersion curves for different longitudinal velocity ratio cL /cs (the higher order branch only exists when cL /cs >1. upper curves: generalized longitudinal waves; lower curves: generalized Rayleigh waves).

an electro-elastic film, which reduces this complex problem to the non-trivial solution of a series of quadratic equations of wave numbers. This model is validated by comparing with exact results, indicating that the model is reliable when the thickness of the layer is smaller or comparable to the typical wavelength. The dispersion curves for specific examples are presented and analyzed. The physical meaning of the two branches of the dispersion curves is examined by analyzing the effect of the transverse inertia and the density of the piezoelectric layer. The influences of the piezoelectric effect and the material combination are also examined and discussed. The piezoelectric effect shows significant influence on the generalized longitudinal wave but only a limited effect on the generalized Rayleigh wave. The upper mode exists only when the longitudinal wave velocity of the substrate is faster than that of the piezoelectric layer. The current model can be used as a benchmark for the study of wave propagation in this type of piezoelectric coupled structures. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by National Natural Science Foundation of China under Grant 51905537, China Scholarship Council and the AITF Top-up award. Thanks to Dr. Cindy Chopoidalo for her proofreading. Appendix A The constitutive equations of piezoelectric materials are given by [30]

⎡

σxx σyy σzz σyz σxz σxy

⎢ ⎢ ⎢ [ ] ⎢ ⎢ σ =⎢ ⎢ D ⎢ ⎢ ⎢ Dx ⎣ D

y

⎤

c11 ⎢ c12 ⎢ c ⎢ 13

⎡

⎥ ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

Dz

[ =

c eT

−λ

][

c13 c13 c33

e31 e31 e33 c44

) 1( ui,j + uj,i , 2

e15 c66

e15 e15

ε −E

e31

e33

]

where

εij =

e15 c44

e31 e

c12 c11 c13

Ei = −V,i ,

i , j = x, y , z

−λ11

− λ11

− λ33

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

εxx εyy εzz 2εyz 2εxz 2εxy −Ex −Ey −Ez

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

H. Yu and X. Wang / Wave Motion 96 (2020) 102559

13

In the rectangular Cartesian coordinates, the equations of motion without body forces are given by

σji,j = ρ u¨ i and Gauss’ law Di,i = 0 In these equations, σij , εij and ui are the stress, the strain and the mechanical displacement, while Di , Ei and V represent the electric displacement, the electric field intensity and the potential, respectively. cij are the stiffness parameters for a constant electric potential, eij are the piezoelectric constants, λij are the dielectric constants for zero strains, and ρ is the mass density. Then, the governing equations of the piezoelectric material can be obtained by combining these equations together, as c11 ux,xx + c66 ux,yy + c44 ux,zz + (c12 + c66 ) uy,xy + (c13 + c44 ) uz ,xz + (e31 + e15 ) V,xz = ρ u¨ x (c12 + c66 ) ux,yx + c66 uy,xx + c11 uy,yy + c44 uy,zz + (c13 + c44 ) uz ,yz + (e31 + e15 ) V,yz = ρ u¨ y (c13 + c44 ) ux,zx + (c13 + c44 ) uy,zy + c44 uz ,xx + c44 uz ,yy + c33 uz ,zz + e15 V,xx + e15 V,yy + e33 V,zz = ρ u¨ z (e31 + e15 ) ux,xz + (e31 + e15 ) uy,yz + e15 uz ,xx + e15 uz ,yy + e33 uz ,zz − λ11 V,xx − λ11 V,yy − λ33 V,zz = 0 It should be noted that these equations are derived based on linear elastic piezoelectricity, and those nonlinear theories need to be considered if displacements become large [31]. For a state of plane strain parallel to the y-z plane, ux,yx = uy,xx = ux,zx = uz ,xx = ux,xz = uz ,xx = 0 Considering the open-loop condition Dy = 0 and Ey = 0, the governing equations under the plane strain (y-z plane) and open-loop conditions can be reduced to 2 ( ∗ ) ∂ 2 uz ∂ 2 uy ∗ ∂ uy ∗ + c44 + c13 + c44 = −ρω2 uy 2 2 ∂y ∂z ∂ y∂ z 2 2 ( ∗ ) ∂ 2 uy ∗ ∂ uz ∗ ∂ uz ∗ c44 + c + c + c = −ρω2 uz 33 13 44 ∂ y2 ∂ z2 ∂ y∂ z

∗ c11

with ∗ ∗ = c13 + e31 e33 /λ33 = c11 + e231 /λ33 , c13 c11 ∗ ∗ = c44 + e215 /λ11 = c33 + e233 /λ33 , c44 c33

Appendix B According to the electro-elastic film model of piezoelectric sensors [29], Ey is zero because the potential is constant along the top and bottom surfaces because of the piezoelectric layer with electrodes. Therefore, the general constitutive relation becomes

σyy = c11 εyy + c13 εzz − e31 Ez σzz = c13 εyy + c33 εzz − e33 Ez Dz = e31 εyy + e33 εzz + λ33 Ez The piezoelectric layer model is basically a beam model with negligible bending stiffness. So similar to Bernoulli– Euler beam, the) normal stress in z-direction inside the beam is small and assumed zero, therefore we have εzz = ( −c13 εyy + e33 Ez /c33 . Then, the stress component σyy and the electric displacement Dz under plane strain condition can be obtained as,

σyy = E

Dz = e

∂ uy − eEz ∂y

∂ uy + λ Ez ∂y

with E = c11 −

2 c13

c33

,

e = e13 − e33

c13 c33

,

λ = λ33 +

e233 c33

When the electric displacement across the piezoelectric layer Dz = 0, the stress component σyy can be determined as

σyy = Eeff

∂ uy , ∂y

with

Eeff = E +

e2

λ

14

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