On propagation behavior of shear wave in piezoelectric-sandwiched structure

1. Introduction

In the past decade, researchers have shown significant interest in the exploration of wave propagation through piezoelectric-layered structures. This heightened attention is attributed to the intriguing applications of such studies in the manufacturing of seismic devices. Piezoelectric materials exhibit a crucial characteristic: a robust correlation between their electric and mechanical behavior. This inherent coupling renders these materials highly valuable in various scientific and technological applications, including displacement transducers, micro-positioners and sensors. Conventional homogeneous piezoelectric materials have some intrinsic defects in cases when load is applied or in terms of lack of stability. To overcome these defects, researchers have introduced the concepts of layered piezoelectric structure, functionally graded piezoelectric materials (FGPM), etc. These advanced piezoelectric structures have improved, durability properties in terms of resistance to deformation, durability and avoidance to stress concentration. To address these limitations, porosity is incorporated into piezoelectric materials, resulting in a reduction of material density and making them well suited for the mentioned applications. This novel category of materials is referred to as porous piezoelectric materials, exhibiting unique properties not achievable by their traditional, dense counterparts. Numerous researchers have explored porous piezoelectric ceramics in previous studies. In his research work, Dunn and Taya¹ utilized a theoretical approach to anticipate the electro-mechanical characteristics of these ceramics. Various scholars examined the characteristics of porous piezoelectric ceramics through experimental approaches.^2,3,4 Subsequently, Vashishth and Gupta⁵ formulated the fundamental constitutive equations and equations of motion for anisotropic porous piezoelectric materials employing a variational principle. In their study, Vashishth and Dahiya⁶ explored the transmission characteristics of SH-type waves within a periodically layered composite structure. This composite structure comprised alternating polymeric layers and porous piezoelectric layers. Additionally, Vashishth et al.⁷ employed the transfer matrix method to examine the propagation of BG-type waves in a layered structure composed of piezoceramic layers, featuring multiple layers and interfacing with a half-space of porous piezoelectric media (PPM). In their study, Sahu and colleagues⁸ examined the transference of Love-type waves within a three-layered structure. This structure consisted of an FGPM layer sandwiched between a piezoelectric layer and a piezomagnetic half-space, all with perfect interfaces. Also, Jiao et al.⁹ studied wave propagation in nonhomogeneous sandwiched structures between piezoelectric and piezomagnetic half-spaces to find the energy reflection and transmission coefficients of various waves. In his paper, Sahu et al.¹⁰ studied the propagation of shear horizontal (SH) surface waves in a composite structure in which an FGP plate fused with a porous piezoelectric plate to study properties for efficient surface acoustic wave (SAW) devices. Kumar et al.¹¹ employed the Wentzel–Kramers–Brillouin asymptotic approach to examine the propagation of horizontally polarized shear (SH) waves in a piezoelectric composite structure. Studies have been also done to study surface wave propagation in piezo-poroelastic structures. In his research work, Sharma¹² studied the propagation of surface waves along all directions on the plane boundary of piezo-poroelastic half-space with arbitrary anisotropy. Many studies have been undertaken about the propagation of Love and shear wave in layered piezoelectric composite structure to discuss the propagation behavior in the medium in terms of porosity and wave number on mechanical displacement, electrical displacement and stress.^13,14,15

Monitoring recent progress and cutting-edge research in the field of sensors and waveguides exposes a discernible gap, prompting an exploration of wave propagation phenomena in layered piezoelectric structures. Incorporating a controlled porosity serves to uphold the acoustic impedance balance between the medium and its surroundings, ultimately resulting in heightened sensitivity of the device. Consequently, investigating wave propagation in layered piezoelectric structures can provide valuable insights into optimizing the implementation of sandwiched structures in diverse sensors and actuators for increased efficiency and thereby enhancing the sensitivity of the device.

The objective of the work is to study the shear wave propagation in a layered piezo-porous structure. The mathematical model is solved analytically to investigate the propagation behavior of shear waves under assumed constraints. The frequency equation has been derived using Biot’s theory of poroelasticity and numerical examples are presented to show the features of dispersion curves for different values of piezoelectricity, anisotropy, porosity, tensile initial stress and thickness of the layer.

2. Statement and Formulation of the Problem

The piezoelectric-layered structure is shown in Fig. 1. The focus of our study is on the transmission of transverse shear waves within layer of transversely isotropic porous material, characterized by its thickness $h_1$ constrained between a piezoelectric layer of thickness $h_3$ and an elastic layer under tensile initial stress of thickness $h_2$, as in shown in Fig. 1. The arrange framework $O–xyz$ is taken such that the z-axis indicating vertically descending. The x-direction is aligned with the propagation direction of the wave.

2.1. Constitutive equations

Field equations for the piezoelectric layer can be expressed as¹⁶

where $i,j=1,2,3$, ${\rho }_1$ denotes mass density, $D_i$ and $u_i$ denote displacements in the ith-axis direction both electrical and mechanical in nature, respectively, ${\sigma }_{ij}$ is the stress. We have followed the Voigt summation convention with respect to time and space coordinates.

The expression of the piezoelectric constitutive equation is as follows :

where ${\sigma }_{ij}$ and $S_{kl}$ are the stress and strain tensor, $D_j$ and $E_k$ are the electrical displacement and electric field intensity and $c_{ijkl}$, $e_{kij}$, ${\epsilon }_{jk}$ are the elastic, piezoelectric and dielectric (tensor notations) coefficients, respectively. Mechanical displacement and the strain components of the medium are related as As per the quasi-static Maxwell’s equation, there exists a connection between the electric field intensity and the electric potential $\varphi$ as The constitutive equations (2) for a transversely isotropic piezoelectric medium with z-axis to be the axis of symmetry¹⁷ can be written as where ${\sigma }_x={\sigma }_{11}$, ${\sigma }_y={\sigma }_{22}$, ${\sigma }_z={\sigma }_{33}$, ${\sigma }_{xy}={\sigma }_{12}$, ${\sigma }_{yz}={\sigma }_{23}$, ${\sigma }_{xz}={\sigma }_{13}$.

Biot^18,19,20 formulated the constitutive equations governing the transversely isotropic porous layer by expressing them in relation to the displacement components of the bulk material $u_i$ and pore fluid displacements $U_i$ ($i=x,y,z$)

The above equation involves the stress tensor (${\sigma }_{ij}$), fluid volumetric strain ($V=-\nabla U_i$), stress acting on the fluid phase ${\sigma }_2(=-f{p}_2)$, fluid pressure $p_2$ and porosity f of the porous layer. The material constants $A_2$, $C_2$, $F_2$, $G_2$, $M_2$, $N_2$, $Q_2$ and $P_2$ are also part of this equation.

The equation that govern shear wave propagation in homogeneous layer with uniform properties in all direction subjected to tensile initial stress is as follows :

where $i=1,2,3$, $P_3$ is the tensile initial stress along the x-direction, ${\mu }_3$ and ${\rho }_3$ is the shear modulus and density of the elastic layer.

The displacement component and the electric potential for each layer are

Substituting Eqs. (8) into Eqs. (4) and (5) and following Eq. (2), the governing equation for the piezoelectric medium can be written as Upon replacing Eq. (6) into equations governing the porous layer without considering body forces and fluid viscosity, following Biot’s work,^18,19,20 we arrive at the following result : The total mass of the fluid–solid aggregate is Also, the dynamic coefficients, moreover, obey the inequalities Using the Love wave conditions (8), Eqs. (7) and (10) reduced to and Solving Eq. (12), we obtain where $m^{\prime }={\rho }_{11}-\frac{{\rho }_{12}^2}{{\rho }_{22}}.$

The piezoelectric layer’s upper surface is exposed to air. As air dielectric constant is significantly lower than piezoelectric material, we consider air to be vacuum for practical purposes. Consequently, the electric potential function ${\varphi }_0$ can be determined using the Laplacian equation

2.2. Boundary conditions

The boundary conditions can be described as follows.

The state of being electrically and mechanically open at the free surface can be expressed as follows :

The expression for the conditions of being mechanically and electrically short at the free surface is as follows : The stress, mechanical displacement, stress, electrical displacement and electrical potential exhibit continuity at the interfaces connecting the piezoelectric layer with the poroelastic medium and with the poroelastic medium to elastic bottom layer as

2.3. Solution of the problem

From Eq. (14) and considering the boundary condition $z\to -\infty$, ${\varphi }_0\to 0$, one can easily obtain the solution of the electrical potential function in vacuum as

The solution of Eq. (13) can be assumed as where k is the wave number, $\omega$ is the angular frequency and c is the common wave velocity.

By substituting Eq. (19) into Eq. (13), we obtain

or where $a_1^2=\gamma d(\frac{c^2}{{\beta }^2}-\frac{1}{d})$, $\gamma =\frac{N_2}{G_2}$ and ${\beta }^{\prime }=\sqrt{\frac{N_2}{m^{\prime }}}$ is the velocity of shear wave in porous layer. The shear wave velocity ${\beta }^{{}^{\prime }}$ can be also written as ${\beta }^{\prime }={\beta }_a\sqrt{\frac{1}{d}}$, where $d={\eta }_{11}-\frac{{\eta }_{12}^2}{{\eta }_{22}}$ and $\beta =\sqrt{\frac{N_2}{{\rho }^{\prime }}}$ is the shear wave velocity in the x-direction for a anisotropic nonporous elastic medium and the dimensionless parameters for the poroelastic layer are given by ${\eta }_{11}=\frac{{\rho }_{11}}{p^{\prime }}$, ${\eta }_{12}=\frac{{\rho }_{12}}{p^{\prime }}$ and ${\eta }_{22}=\frac{{\rho }_{22}}{p^{\prime }}$.

Therefore, the solution for the displacement component $u_y$ is

To write Eq. (9) explicitly in $u_y^1$ and $\varphi$, we assume

which reduces Eq. (9) to where $c_{\text{sh}}^2=\frac{1}{{\rho }_1}(c_{44}+\frac{e_{15}^2}{{\epsilon }_{11}})$.

The wave propagation solution for $u_y^n(n=1,2)$ and $\psi$ in x-direction can be formulated as

Substituting Eq. (25) in Eqs. (23), (24) and (11), we obtain

Using Eqs. (26), solution for $u_y^1$, $\phi$ and $u_y^2$ from can be written as where $b_1=\sqrt{\frac{c^2}{c_{\text{sh}}^2}-1}$, $b_2=\sqrt{\frac{P_3}{2{\mu }_3}+\frac{c^2}{{\beta }_3^2}-1}$ and ${\beta }_3=\sqrt{\frac{{\mu }_3}{{\rho }_3}}$ are velocities of shear wave in the elastic layer with tensile initial stress.

3. Shear Wave Propagation and Dispersion Relation

By utilizing the boundary conditions in Eq. (15) and continuity conditions described in Eq. (17) concerning the electrically open scenario, we derived a collection of homogeneous algebraic equations aimed at finding the unknown constants $c_i$ ($i=0-8$).

In the electrically open case scenario, we can derive the algebraic equation for the unknown constants $c_{i}i=0$–8 by substituting the solution for $u_y^i$ ($i=1$, 2), $u_y$ and $\varphi$ and the boundary conditions mentioned in Eqs. (15) and (17). This results in the following set of algebraic equations :

which represent the frequency equation for Love wave. For $c>c_{\mathrm{\text{sh}}}$ ($c_{\mathrm{\text{sh}}}$ denotes velocity of shear waves in upper layer), we have All others term of Eq. (30) are equal to zero.

In the electrically short case scenario, we can derive the algebraic equation for the unknown constants $c_{i}j$ = 1–8 ($c_0$ are additional constant) by substituting the solution for $u_y^i$ ($i=$1, 2), $u_y$ and $\varphi$ and the boundary conditions mentioned in Eqs. (15) and (17). This results in the following set of algebraic equations :

Equation (31) represents the dispersion relation connecting the wave number and phase velocity in the scenarios of electrical open and electrical short cases can be derived from Eqs. (30) and (31).

4. Numerical Data

The propagation characteristics of shear wave based on dispersion relation in Eqs. (30) and (31) are investigated numerically with various selected elastic parameters to demonstrate the dispersion characteristics of the shear wave with $(k_1=h_2∕h_1)$, $(k_2=h_3∕h_1)$, dimensionless tensile stress $\alpha =\frac{P_3}{2{\mu }_3}$ and $\frac{c_{44}}{G}=3.0$, $\frac{c_{44}}{G}=2.0$.

The material constants of the piezoelectric upper layer (PZT-4)²¹ are listed in Table 1.

4.1. Discussion: Dispersion curves in electrical open case

In order to provide comparison of the dispersion characteristics, Figs. 2 and 3 discuss the dimensionless phase velocity against dimensionless wave number at different thickness ratio of the sandwiched poroelastic layer to the upper piezoelectric layer and, to the lower elastic layer of the medium in electrical open case. The curves show that the propagation of shear wave is influenced by thickness ratio $k_1$ for different values of $k_2$. Also, it is evident that when the fixed anisotropy and porosity are held constant, an increase in thickness ratio $k_1$ or $k_2$ results in a decrease in the phase velocity of the shear waves.

Figure 4 has been plotted for dimensionless phase velocity against dimensionless wave number to discuss the effect of porosity of the medium for different values of thickness ratio $k_1$ and $k_2$. It clearly appears that as the porosity increases (i.e., as the value of d decreases) for fixed anisotropy, the phase velocity of shear waves increases.

Figure 5 shows the dispersion curves showing variation of phase velocity against dimensionless wave number for different values of piezoelectricity with different values of thickness ratio parameter $k_1$ and $k_2$. It can be seen that for fixed anisotropy and porosity and with varying thickness of layers disfavors the phase velocity of shear wave. However, when the thickness of each layers is same piezoelectricity favors phase velocity so that the phase velocity increases with the increase of piezoelectricity.

Anisotropy of sandwiched structure has a significant effect on the dispersion characteristics of shear wave in electrically open case. Figure 6 shows how anisotropy of a medium going to effect the phase velocity of shear waves with different values of thickness ratio $k_1$ and $k_2$ in the layered structure. It can be seen that for fixed value of porosity, the anisotropy of the medium decreases the phase velocity.

Figure 7 illustrates the impact of tensile initial stress on the medium. It is evident that, with a constant anisotropy and porosity, the phase velocity of the medium decreases as the tensile initial stress increases. It is also evident that when the thickness of all layers is same, Phase velocity is different for lower wave number and converges for higher wave number.

4.2. Dispersion curves in electrical short case

Figures 8 and 9 show the variation of phase velocity for the effect of porosity and anisotropy. As the porosity increases while keeping the anisotropy fixed, the phase velocity of shear waves also increases. On the other hand, when the anisotropy of the poroelastic layer increases, the phase velocity of shear waves decreases. Nonetheless, with an increase in wave number, all the curves converge.

For electrically short case, Fig. 10 shows the dispersion curves showing the variation of phase velocity against dimensionless wave number for different values of piezoelectricity with different values of thickness ratio parameter $k_1$ and $k_2$. It can be seen that for fixed anisotropy and porosity and with varying thickness of layers, piezoelectricity of the medium favors the phase velocity of shear wave.

5. Conclusions

Large number of studied have been done to highlight the utility of piezoelectric material in umber of application. Also, Wave propagation in piezoelectric media have been studied in connection with the generation and transmission of disturbances in electro-acoustic devices such as SAW devices, resonators and transducers. The study of coupling between electrical, mechanical and piezoelectric effects of piezoelectric-porous system can done using the piezoelectric effects in piezoelectric-porous ceramics. So, the study of propagation of SH-waves considered here is very important. The velocity profile of SH-waves is analyzed using an analytical approach, taking into account the impact of anisotropy, porosity, piezoelectricity and initial stress. The study also examines the influence of geometry on each of these parameters, demonstrating the results explicitly. Frequency equations of the wave propagation has been derived, MATLAB and MATHEMATICA have been used for graphical interpretation in both electrical open and short case. We may conclude with the following results:

(1)	Dimensionless phase velocity increases with the decrease of dimensionless wave number in both electrical open and short case. Also, phase velocity varies for different values of thickness ratio of the layers.
(2)	As the wavelength of SH waves is smaller than the thickness of piezoelectric layer at higher wave number, the piezoelectric layer dominates the dispersion of the SH wave propagation.
(3)	It is observed that the porosity of the medium tends to increase the phase velocity of the shear waves in electrical open and short case.
(4)	Increment in anisotropy and tensile initial stress of the medium decreases the phase velocity of shear waves for different thickness ratio in both electrical open and short cases.
(5)	It is observed that when the thickness of the piezoelectric layer is greater in comparison to other layers, the phase velocity of shear waves decreases with the increase of piezoelectric parameter.
(6)	It is also observed when the thickness of all the layers is same, phase velocity of shear waves increases with the increases of piezoelectric parameters.

Obtained results give a theoretical guidance for shear wave propagation in piezo-porous-layered structure and can be used for studies aimed to improving the designs of SAW devices used in signal processing and communication. The research holds the potential to aid in the creation of enhanced sensor and acoustic devices. This can be achieved by carefully choosing suitable materials, elastic constants, thicknesses and different boundary conditions.

ORCID

Pradeep Kumar Saroj   https://orcid.org/0000-0001-9704-7582