Nomenclature

c_ijkl, elastic constant of the functionally graded material (FGM) layer;

ε_ij, dielectric constant of the FGM layer;

ρ^f, mass density of the FGM layer;

D_i, electric displacement vector in the FGM layer;

α, gradient factor of the FGM layer;

σ _ij' (σ^*), stress component in the porous piezoelectric (PP) medium for the solid (fluid) phase;

s_ij' (s^*), strain component in the PP medium for the solid (fluid) phase;

σ_kj⁰, initial stress tensor;

c_ijkl', elastic constant of the PP medium;

D_i' (D_i^*), electric displacement vector in the PP medium for the solid (fluid) phase;

E_i' (E_i^*), electric field vector in the PP medium for the solid (fluid) phase;

m_ij, elastic constant of the PP medium;

R, elastic constant of the PP medium;

e_ijkl', e_i^*, ς_ijk, ς_i, piezoelectric constants of the PP medium;

ε _ij', ε_ij^*, A_ij, dielectric constants of the PP medium;

φ^p (φ^*), electric potential function in the PP medium for the solid (fluid) phase.

1 Introduction

Piezoelectric materials are smart materials capable of producing electrical fields under mechanical stress. When such materials are embedded in composite structures, smart structures will form. Piezoelectric materials have extensive applications, e.g., actuators, sensors, surface acoustic wave (SAW) devices, and transducers^[1-2]. Despite numerous applications, the smart structures consisting of piezoelectric materials have many shortcomings. Piezoelectric materials are brittle, which may cause failure of piezoelectric material devices under electrical and mechanical loading. Pieoelectric materials cannot be utilized in undersea applications since they have high acoustic impedance. These limitations can be overcome by introducing porosity in a controlled manner to the piezoelectric material (hence reducing the material density). The piezoelectric materials containing tailored porosity are called porous piezoelectric materials. They show specific attributes that cannot be exhibited by their regular dense analogues. Such materials have tremendous utilization in ultrasonic transducers, hydrophones, etc.

Functionally graded materials (FGMs) are a class of materials made of more than one constituent phases. In FGMs, the mechanical properties, i.e., Poisson's ratio, Young's modulus, shear modulus of elasticity, and density, vary continuously and smoothly in the preferred direction^[3]. FGMs have wide applications in solar cells, spacecraft heat shields, biomedical implants, thermoelectric generators, heat exchange tubers, sensors, etc. They can also be used in the medical field for dentistry, skins, and artificial bones^[4].

Since the material properties of the FGM layer is non-uniform and the thermal expansion, chemical shrinkage, etc. are prominent, the presence of initial stresses is unavoidable in an FGM layered structure. Moreover, the layered structure is pre-stressed to prevent from brittle fracture. The initial stresses in the layered structures can lead to delamination, microcracking, debonding, and degradation of the layer. Therefore, the effects of the initial stresses on the existence and propagation behavior of transverse surface waves in such layered structures are important to be investigated.

The compositions of FGMs and porous piezoelectric ceramics can reduce the brittleness of the composite structures, which leads to the enhancement of the strength and results in the increase in the performance of the devices. In recent years, the study of shear wave propagation in FGM composite structures has turned into a focus point of many researchers. Han et al.^[5] studied the transmission of elastic waves in FGM plates with a numerical method. They divided the FGM layer into quadratic layer elements (QELs), and presented a characterization of the material property of the FGM. Han and Liu^[6] used a computational technique to analyze the propagation of shear waves in FGM plates. Qian et al.^[7] obtained the analytical solutions for the transference of Love-type waves in a structure consisting of an FGM layer lying over the piezoelectric half-space with the Wentzel-Kramers-Brillouin (WKB) asymptotic technique. Zhang and Batra^[8] studied the transmission of elastic waves in FGMs with a modified smoothed particle hydrodynamics method. Bin et al.^[9] used the Legendre orthogonal polynomial series expansion approach to explore the propagation of harmonic waves in functionally graded magneto-electro-elastic plates. Qian et al.^[10] used the WKB technique to study the Love-type wave propagation in the FGM half-space. Aksoy and Şnocak^[11] adopted the space-time discontinuous Galerkin method to examine the wave propagation in layered materials. Ravasoo^[12] studied the counter-propagation of two ultrasonic harmonic waves in an inhomogeneous material having exponentially varying physical properties. Cao et al.^[13] analyzed the effects of the gradient coefficient on the Lamb wave propagation in an FGM plate with the power series method. Kiełzyńki et al.^[14] investigated ultrasonic Love-type waves in the FGMs having inhomogeneity. Arani et al.^[15] studied the effects of the material in-homogeneity on the electro-thermo-mechanical behaviors of a rotating hollow circular shaft made from functionally graded piezoelectric material (FGPM). Sahu et al.^[16] studied the shear wave propagation in a composite structure consisting of an FGPM layer bonded between a piezomagnetic layer and the elastic half-space. Singhal et al.^[17] investigated the transmission of the surface waves in an FGPM layer sandwiched between a piezomagnetic (PM) layer and a piezoelectric (PE) substrate having initial stress. Mondal and Sahu^[18] studied the propagation of shear horizontal (SH) waves in a corrugated FGPM layer lying over a piezomagnetic half-space.

Remarkable works have been performed addressing the propagation of waves in composites consisting of porous piezoelectric materials. Craciun et al.^[19] investigated the propagation of elastic waves in porous piezoelectric ceramics. The influence of porosity on the properties of porous piezoelectric materials has been experimentally studied by many authors^[20-22]. Kar-Gupta and Venkatesh^[23] used the finite element method to study the effects of porosity on piezoelectric materials. Vashishth and Gupta^[24] derived the constitutive and dynamical equations for porous piezoelectric materials. Vashishth et al.^[25] investigated the propagation of a Bleustein-Gulyaev (B-G) type wave in a layered porous piezoceramic structure. Gaur and Rana^[26] analytically studied the SH wave propagation in a two-layered composite structure comprising porous piezoelectric and piezoelectric materials. Vashishth and Gupta^[27] studied the dispersion of waves in a porous piezoelectric (PP) medium with the Christoffel equation. Baroi et al.^[28] studied the propagation of SH waves in a viscous liquid overlying a porous piezoelectric half-space. Despite these works, the wave propagation in a PP layer still needs more investigation. There are certain characteristics of PP materials revealed to get contrast findings in wave propagation phenomena.

In this paper, the propagation behaviors of Love-type waves in a structure consisting of an FGM layer and a porous piezoelectric substrate are studied. The FGM layer is assumed to have constant initial stresses. For the solution of the FGM layer, the WKB approximation method is used. For the solution of a porous piezoelectric medium, the variable separable method is used. Both electrically open and short cases are considered. The dispersion relation is obtained in the determinant form for each case. Numerical examples are given to exhibit the analytical results graphically. The effects of the material gradient of the FGM layer and the initial stresses (tensile and compressive stresses) on the velocity profile of the Love-type wave are shown clearly. The dependence of height of the FGM layer on the velocity of the considered wave is illustrated. The obtained results can be utilized for the improvement of piezoelectric devices.

2 Formulation of the problem

We contemplate the Love-type wave propagation in a bedded structure comprising of an FGM layer and a porous piezoelectric substrate (see Fig. 1). In the Cartesian coordinate system, the x-axis lies vertically down and the y-axis extends in the wave propagation direction. We assume that the FGM layer has constant initial stresses and the surface is traction-free. Usually, for SAW devices, the thickness of the substrate is much larger than that of the layer. Therefore, the structure may be treated as a layer-half-space problem.

2.1 Formulation for the FGM layer

For the wave motion with small amplitude, the field equations can be expressed as follows^[10]:

(1)

and the constitutive relations are as follows:

(2)

where i, j, k=1, 2, 3, and c_ijkl and ε_ij are elastic and dielectric constants, respectively. ρ is the mass density. u_i denotes the mechanical displacement in the ith-direction. D_i denotes the electric displacement vector. σ_ij and σ_kj⁰ denote the stress tensor and the initial stress tensor, respectively.

The strain tensor is related to the displacement by , and the electric field is related to the electric potential by

(3)

Considering the wave propagation in the y-axis, the mechanical displacement components and the scalar electric potential function become

(4)

Substituting Eq. (4) into Eq. (1) yields

(5)

(6)

where w^f, ρ^f, c₄₄^f, and φ ^f denote the mechanical displacement, the mass density, the shear modulus, and the electric potential in the FGM layer. The mass density is assumed to be constant, while the shear modulus of the FGM layer is considered to be varying exponentially in the thickness direction.

We assume that g(x)=e^αx, c₄₄⁰ =c₄₄(0), and ρ^f is constant, where α is the gradient coefficient.

Substituting Eqs. (2), (3), (4) into Eq. (1), from Eqs. (5) and (6), we have

(7a)

(7b)

Also, we have the components of σ_ij and D_i as follows:

(8)

where represents the shear wave velocity at the x=0 interface.

2.2 Formulation for the PP medium

For the PP medium, the equations of motion are as follows^[25]:

(9)

where u_i and U_i^* are the mechanical displacement components of M₂, and (ρ₁₁)_ij, (ρ₁₂)_ij, and (ρ₂₂)_ij are the dynamical mass coefficients.

The constitutive equations for the PP medium are

(10)

where σ_ij' (σ^*) is the stress tensor component for the solid (fluid) phase. s_ij'(s^*) is the strain tensor component for the solid (fluid) phase. D_i' (D_i^*) is the electric displacement for the solid (fluid) phase. E_i' (E_i^*) is the electric field vector for the solid (fluid) phase. c_ijkl', m_ij, and R are the elastic constants. e_ijkl', e_i^*, ς_ijk, and ς_i are the piezoelectric constants. ε _ij', ε_ij^*, and A_ij are the dielectric constants. Let φ^p (φ^*) be the electric potential function for the solid (fluid) phase.

Then, we have

(11)

Assume that the waves propagate in the y-axis. Then, the mechanical displacements and the electric potential functions are

(12)

and Eq. (10) can be rewritten as follows:

(13)

Substituting Eqs. (11), (12), and (13) into Eq. (9) yields

(14)

(15)

where

(16)

3 Solution of the problem 3.1 Solution for the FGM layer

We assume the solution of the displacement component for the FGM as follows:

(17)

Substituting Eq. (17) into Eq. (7) yields

(18)

We introduce the following transformation:

(19)

Now, Eq. (18) is converted to the following form:

(20)

where g' and ψ' are derivatives of g and ψ w.r.t. x.

To solve Eq. (20), which is a non-linear differential equation in ψ, we expand ψ as follows:

(21)

Substituting Eq. (21) into Eq. (20) yields

(22)

Then, we get the following equations when the coefficients of each power of k are equated to zero:

(23)

Let g(x)=e^{α x}. Then, Eq. (23) gives the solutions of ψ₀, ψ₁, ψ₂, ... as follows:

(24)

where

Now, we take two solutions (ψ₀, ψ₁), and substitute them into Eq. (21). Then, using Eq. (19), we get the solution of Eq. (18) as follows:

(25)

where A₁ and A₂ are unknown constants.

Substituting Eq. (25) into Eq. (17) yields the solution of the mechanical displacement in the FGM layer as follows:

(26)

Solving Eq. (7b), we obtain the electric potential function in the layer as follows:

(27)

3.2 Solution for the PP substrate

We assume the solution of the mechanical displacement component and the electric potential function for the porous piezoelectric material as follows:

(28a)

(28b)

Putting Eqs. (28a) and (28b) into Eqs. (14) and (15), we get

(29a)

(29b)

Then, we have

where

Here, β₂ represents the bulk shear wave velocity in the PP substrate.

For x→ +∞, w^p→ 0, and φ^p→ 0, the solution of the mechanical displacement and the electric potential function in the porous piezoelectic half-space is obtained as follows:

(30)

(31)

4 Boundary conditions and dispersion relations

(ⅰ) When the surface of the FGM layer is electrically open,

(32)

(ⅱ) When the surface of the FGM layer is electrically short,

(33)

(ⅲ) The continuity conditions at the interface of the FGM layer and the porous piezoelectric substrate, i.e, at x=0, are

(34)

4.1 Dispersion relation for the electrically open case

Using Eqs. (26), (27), (30), and (31) along with the boundary conditions (32) and (34), we obtain the following equations:

(35)

(36)

(37)

(38)

(39)

(40)

where

It can be seen that Eqs. (35)--(40) are 6 equations with 6 variables. For non-trivial solutions, the determinant of the coefficient matrix vanishes, which leads to the dispersion relation as follows:

(41)

The non-zero coefficients of the above determinant equation (41) are provided in Appendix A.

4.2 Dispersion relation for the electrically short case

Using Eqs. (26), (27), (30), and (31) along with the boundary conditions (33) and (34), we obtain the equations the same as the above except Eq. (36), which changes as

(42)

The dispersion relation for the electrically short case is obtained in the determinant form as follows:

(43)

The non-zero coefficients of the above determinant equation (43) are provided in Appendix B.

5 Validation of the problem

Case Ⅰ  When the piezoelectric effect is neglected in the lower medium, the considered model reduces into the FGM layer embedded over the elastic substrate. The dispersion relation for the reduced model is obtained as follows:

(44)

which resembles with the phase velocity equation of the Love-type wave propagation attained by Qian et al.^[10].

Case Ⅱ  When the considered structure is reduced into a homogeneous isotropic layer lying over a homogeneous isotropic semi-infinite medium, the dispersion relation becomes

(45)

where

Equation (45) matches with the classical Love-type wave equation with the condition β'₁ < c < β'₂, which is the required sequence of the Love-type wave velocity in the layered structure^[29].

6 Numerical example and result discussion

The data taken for the numerical purpose are given in Table 1.

An analytical solution is obtained for the transference of Love-type waves in a composite structure consisting of an FGM layer and a PP substrate. The phase velocity of the considered wave satisfies the condition β₁ < c < β₂, where β₁ and β₂ are the bulk wave velocities in the FGM layer and the PP substrate, respectively. The above condition can also be verified through graphs drawn for the dimensionless phase velocity (c/β₁) versus the dimensionless wave number (kh₁). The range of the dimensionless phase velocity agree well with the condition β₁ < c < β₂.

Using the dispersion relations (41) and (43), numerical examples are given. For the numerical computation, PZT-5H is taken as the porous-piezoelectric layer. The phase velocity of the Love-type wave decreases with the wave number.

Figures 2 and 3 give the variations of the dimensionless phase velocity of the Love-type wave with the dimensionless wave number for different values of the gradient factor of the FGM layer for electrically open and short cases, respectively. We notice that, in both cases, the phase velocity gets reduced with the increase in the gradient factor.

Figures 4 and 5 give the variations of the non-dimensional phase velocity of the Love-type waves with the non-dimensional wave number width of the FGM layer for electrically open and short cases, respectively. It is clearly visible that the phase velocity decreases when the width of the FGM layer increases.

The effects of the initial stresses on the phase velocity of the Love-type wave in the FGM layer is shown in Figs. 6 and 7 for the tensile stress and the compressive stress, respectively, for the open case.

The effects of the initial stress on the phase velocity of the Love-type wave are shown in Figs. 8 and 9 for the tensile stress and the compressive stress, respectively, for the short case.

It is seen that the initial stress has slight effect on the phase velocity of the Love-type wave. It is noticed that the effect of the initial stress is negligible when μ₀ < 10⁸ Pa. When μ₀ >10⁸ Pa, the phase velocity increases with the increase in the tensile stress whereas decreases with the increase in the absolute value of the initial compressive stress.

7 Conclusions

The propagation of Love-type waves in a bedded structure comprising of an FGM layer with a constant initial stress lying over a porous-piezoelectric substrate is taken into consideration. The WKB approximation technique and the variable separable method are applied to obtain the expressions for the mechanical displacement and the electric potential function of the FGM layer and the PP substrate, respectively. Dispersion relations are obtained with the consideration of the appropriate boundary conditions for two cases, i.e., electrically open and electrically short. For the numerical illustrations, the material constants of PZT-5H are taken for the porous-piezoelectric layer. We summarize with the following prominent observations:

(ⅰ) The material gradient of the FGM layer has a significant effect on the phase velocity of Love-type waves. The phase velocity of the considered wave decreases when the gradient factor of the FGM layer increases in both electrically open and short cases.

(ⅱ) The phase velocity of Love-type waves decreases with the increase in the width of the FGM layer in both electrically open and short cases.

(ⅲ) The effect of the initial stress is negligible when | μ₀| < 10⁸ Pa. The phase velocity of the Love-type wave increases with the increase in the initial tensile stress whereas decreases with the increase in the absolute magnitude of the initial compressive stress when |μ₀|>10⁸ Pa.

(ⅳ) The present study contributes towards the design and optimization of underwater acoustic devices.

Appendix A Appendix B