Document Type : Research Article
Authors
^{1} School of Science and Technology, The Neotia University, Sarisha, West Bengal, 743368, India
^{2} Department of Mathematics, Brainware University, Barasat, West Bengal, 700125, India
^{3} Department of Basic Science, MCKV Institute of Engineering, Howrah, West Bengal, 711204, India
^{4} School of Arts and Natural Sciences, Joy University, Raja Nagar, Kanyakumari, Tamilnadu, 627116, India
Abstract
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Main Subjects
Research Article
Suparna Roychowdhury ^{a,b }, Abhijit Pramanik ^{c } , Mostaid Ahmed ^{a,d * }
^{a} School of Science and Technology, The Neotia University, Sarisha, West Bengal, 743368, India
^{b} Department of Mathematics, Brainware University, Barasat, West Bengal, 700125, India
^{c} Department of Basic Science, MCKV Institute of Engineering, Howrah, West Bengal, 711204, India
^{d} School of Arts and Natural Sciences, Joy University, Raja Nagar, Kanyakumari, Tamilnadu, 627116, India
ARTICLE INFO 

ABSTRACT 
Article history: Received: 20230826 Revised: 20240402 Accepted: 20240521 

The article incisively analyzes the impact of piezoelectricity on Love wave transmission in an inhomogeneous bilayered structure consisting of smoothly embedded thin piezoelectric material bonded to a semiinfinite fiberreinforced medium. By applying the variableseparable method, a general form of dispersion equations, analyzing the Love waves’ characteristics in electrically open and shorted cases of the piezoelectric material has been derived. The crux of the study lies in the fact that the presence of the prestresses in the upper layer and lower halfspace along with elastic, piezoelectric, and permittivity coefficients lead the derived frequency relation to merge with the classical form of equations of the Love waves. The procured dispersion relation substantiates that the depth of the upper layer prestresses, and piezoelectricity coefficients play a guiding role in the transmission of Love waves. The numerical discussions and findings carry wider applications and may imply guidance of additive manufacturing of varied composite multimaterials for prestressed and microstructural configurations with partial and global dispersion properties 



Keywords: Love waves; Piezoelectricity; Fiberreinforced; Electrically short case; Electrically open case. 


© 2025 The Author(s). Mechanics of Advanced Composite Structures published by Semnan University Press. This is an open access article under the CCBY 4.0 license. (https://creativecommons.org/licenses/by/4.0/) 
The advent of piezoelectric materials which tend to hold unique characteristics to generate electric charge potentials subject to any form of mechanical stresses comprising compression, bending, and stretching leads to several engineering applications. Piezoelectric materials carry several utilitarian aspects in devices like sensors, actuators, filters, etc. To gain more insights into the localization properties of Love waves at piezoelectric surfaces, several researchers have worked on enhancing the sensitivity of SAW devices. Sensors remain highly sensitive in lieu of acoustic energy accumulation within specified wavelengths. In present times, the problems associated with the reflection and refraction phenomena of elastic waves have attracted paramount interest in the field of waves and acoustics [16]. Moreover, the plane wave transmission in anisotropic media traces some slightly different characteristics from those of inisotropic media. Li and Jin [7]) expressed how the SH waves propagate in piezoelectric materials not smoothly bonded to a metal or elastic substrate. Chen et al. [8] illustrated the electromechanical characteristics of a piezoelectric actuator projected to a graded substrate including an adhesive material is studied with graphical representations. Chen et al. [9] investigated the interfacial imperfections of thin piezoelectric film clamped to a graded substrate. Chen et al. [10] developed the relations for interfacial behavior of a piezoelectric actuator lying over to a homogeneous halfspace. Cooper and Reiss [11] described the incidence of reflection of harmonic plane waves on the free surface of a linearly viscoelastic semiinfinite space. The incidence of reflection and refraction of elastic waves on inhomogeneous earth media was extensively described by Nayfeh [12]. Du et al. [13] introduced how Love waves behave while propagating through functionally graded piezoelectric materials. Du et al. [14], in an analytical investigation, have shown the influence of prestress in piezoelectric layered structures underlying viscous liquid. Love waves involving a thin piezoelectric layer clamped smoothly to unbounded elastic subs. The propagation characteristics of Lovetype waves in semiinfinite FGPM with quadratic variation were addressed by Eskandari and Shodja [15]. Liu and He [16] carried out extensive work to derive a general frequency equation for Love waves transmitting in a layered structure over a piezoelectric halfspace. In addition to these studies, some unparalleled works have been nurtured by many authors discussing the problems relevant to piezoelectric materials and fiberreinforced materials. The expressions for reflection coefficients of the reflected waves in lieu of P and SV waves on an insulated and isothermalfree surface were demonstrated by Othman and Song [17]. Tahir et al.[18,19] established the frequency relations for wave propagation for ceramicmetal functionally graded sandwich plates with different porosity distributions and functionally graded sandwich plates via a simple quasi3D HSDT. AlFurjan et al. [20] studied the fundamental properties of waves in a sandwiched structure with a soft core and multihybrid nanocomposite (MHC) face sheets under higherorder shear deformable theory. Due to varied applications in smart devices, the reflection and transmission phenomena on piezoelectric and other complex materials are represented by several scientists and engineers across the globe. The incidence of reflection and transmission of plane waves at the interface of the piezoelectric and piezomagnetic substrates was analyzed by Pang et al. [21]. Moreover, Yuan and Zhu [22] studied the reflection and refraction of plane waves at an interface of two anisotropic piezoelectric substrates.
In the foundational work on piezoelectric plate vibrations, Tiersten [23] comprehensively explored linear vibrations in piezoelectric plates. This seminal publication by Tiersten in 'Linear Piezoelectric Plate Vibrations' from Plenum Press in New York, NY, significantly contributes to the understanding of piezoelectric plate dynamics.
In the exploration of guided waves, Nie, An, and Liu [24] extensively studied the characteristics of SHguided waves in layered piezoelectric/piezomagnetic plates. In the context of characterizing fiber composites, Markham [25] made significant contributions to the field through the measurement of elastic constants using ultrasonics. This pivotal work by Markham on 'Measurement of the Elastic Constants of Fiber Composites by Ultrasonics' provides a crucial foundation for understanding the mechanical properties of such materials.
A comprehensive analysis of Free vibration, wave propagation, and tension analyses of a sandwich micro/nanorod subjected to electric potential using strain gradient theory was carried out by Arefi, M. and Zenkour, A.M., [26]. Arefi and Zenkour [27] conducted an insightful analysis of the wave propagation behavior of a functionally graded magnetoelectroelastic nanobeam resting on a ViscoPasternak foundation. Their study contributes significantly to understanding the intricate dynamics of such nanostructures. In the study of microscale effects on structural behavior, Arefi and Zenkour [28] extensively investigated the influence of microlengthscale parameters and inhomogeneities on the bending, free vibration, and wave propagation analyses of a functionally graded Timoshenko’s sandwich piezoelectric microbeam. Their research significantly contributes to understanding the nuanced mechanical responses of such microstructures. Further Arefi, M. [29] worked on the Surface effect and nonlocal elasticity in wave propagation of functionally graded piezoelectric nanorod excited to an applied voltage which has extensive implications on Love waves propagating in piezoelectric layer embedded over a fiberreinforced substrate. In recent years, there has been a growing interest in the vibrational behavior of nanostructures, particularly those with piezoelectric properties. Arefi [30] conducted a comprehensive nonlocal free vibration analysis of a doubly curved piezoelectric nanoshell, shedding light on the intricacies of the dynamic response in such structures. This work serves as a foundation for understanding the unique characteristics of piezoelectric nanoshells, providing valuable insights that motivate our exploration into Love waves propagating in piezoelectric layer embedded over a fiberreinforced substrate. As the field of structural analysis continually evolves, recent studies have explored advanced theories for a more accurate understanding of the mechanical behavior of materials. Arefi et al. [31] made notable contributions by applying the sinusoidal shear deformation theory and introducing the concept of a physical neutral surface in the analysis of functionally graded piezoelectric plates. Their work provides a sophisticated framework for investigating the intricate mechanical responses of such composite structures. In the realm of structural dynamics, the free vibration analysis of composite structures has garnered significant attention, particularly in the exploration of advanced materials. Arefi, MohammadRezaei Bidgoli, and Zenkour [32] made a significant contribution by conducting a meticulous study on the free vibration analysis of a sandwich nanoplate. Their work, which includes a consideration of a functionally graded (FG) core and piezoelectric face sheets while accounting for the neutral surface, offers valuable insights into the dynamic behavior of such complex structures.
In the exploration of wave propagation in nanomaterials, Arefi and Zenkour [33] made a significant contribution by employing coupled stress components and surface elasticity in the nonlocal solution of a functionally graded piezoelectric Love nanorod model. This insightful work addresses the complexities associated with wave propagation in nanorods, laying the groundwork for a deeper understanding of the mechanical behavior in such intricate systems. In the realm of elasticity, Behdad and Arefi [34] proposed a mixed twophase stress/straindriven elasticity, presenting applications in static bending, vibration analysis, and wave propagation In the dynamic analysis of nanoscale structures, Behdad and Arefi [35] also had introduced a novel mixture model of elasticity, offering insights into buckling analysis, vibration behavior, and wave propagation in such structures. This pioneering work addresses fundamental aspects of nanoscale dynamics, providing a valuable framework for understanding the mechanical responses of these systems. In the area of plate reinforcement, Belfield, Rogers, and Spencer [36] explored stress distributions in elastic plates reinforced by fibers arranged in concentric circles. In the exploration of wave interactions at material interfaces, Das et al. [37] investigated the reflection and refraction of plane waves specifically at the loosely bonded common interface of piezoelectric fiberreinforced and fiberreinforced composite media. Their study provides valuable insights into the complex phenomena occurring at these interfaces.
In the investigation of sizedependent phenomena in micro/nanostructures, Dehsaraji, Arefi, and Loghman [38] conducted a thorough free vibration analysis of functionally graded piezoelectric micro/nanoshells. Their study, based on the modified couple stress theory and considering thickness stretching effects, contributes significantly to understanding the intricacies of sizedependent behaviors in these structures. Ezzin et al. [39] presented a comprehensive study on the propagation of guided waves in functionally graded magnetoelectro elastic composites. Their work contributes significantly to the understanding of the complex behavior of guided waves in such composite materials.
In the investigation of SHwave propagation in porous piezoelectric composites, Rakshit et al. [40] explored the effect of interfacial imperfections. Their study contributes valuable insights into the complex interplay between interfacial imperfections and wave propagation characteristics in these composite materials. In the context of wave propagation studies, Singh, Mistri, and Das [41] investigated the propagation of Lovetype waves in a corrugated fiberreinforced layer. Their research provides valuable insights into the dynamic behavior of such composite structures and in [42] Singh et al. employed a Green's function approach to study the influence of a point source. Their use of this approach provides a comprehensive understanding of the dynamic behavior of piezoelectric materials subjected to SH waves. A few pathbreaking works on thermomechanical vibrations on piezoelectricity were carried out by Khorshidi and Karimi [43] and Khorshidi and Pagoli [44].
Beneath the earth, there are several pieces of evidence found that fiberreinforced media may also coexist with piezoelectric materials with elevated value of initial stress. The characteristic features of this type of material are different from the integral materials, namely concrete and steel. Due to its reduced weight and high strength, fiberreinforced composites are used in various structural elevations. Inside composite materials, individual materials remain distinct. This material consists of the fiber, the matrix, and the interface between materials. PZT (piezoelectric ceramics) and PVDF (piezoelectric polymers), remained frequently usable in actuators and sensors for monitoring and repairing health materials. The authors have applied the separation of the variable method to analyze the dispersion relation in order to characterize the properties of the Love type waves in the aforementioned model and are validated with numerical results.
In this model, we have considered two media involving a piezoelectric layer and fiberreinforced halfspace under initial stresses respectively. Fiberreinforced halfspace is rested beneath a piezoelectric layer consisting of thickness h (in meters) (where ). The geometry of the model is tailored in such a way that the Love wave is transmitted along the xaxis and the zaxis is pointing vertically downwards as shown in Figure 1. The polarization direction of piezoelectric material is assumed to be along the direction of the yaxis.
Fig. 1. Geometry of the problem
The governing equations of piezoelectricity with prestress in the absence of body forces are given (Tiersten, [23])

(1) 
in which indicates stress tensor and is the electric displacement, is the density of the material, and indicates the prestress. The subscript ‘,’ and a superimposed ‘.’ indicate the usual meaning of a partial derivative in respect of the coordinates and derivative with respect to time, respectively.
The governing relations for an anisotropic and linearly electroelastic solid may be represented as

(2) 
where and are the strain tensor and intensity of the electric field respectively. , and indicate the elastic, piezoelectric, and permittivity coefficients, respectively.
The straindisplacement relations are given by:

(3) 
in which is the electric potential component and assuming the prestress P acting along the direction of the xaxis.
Displacement components and the electrical potential function are assumed to be

(4) 
in the case of Love wave transmitting horizontally and in which u, v, and w are the mechanical displacement components along the directions of the x, y, and zaxis respectively.
On Substitution of equation (3) in equation (2) and in equation (1) with consideration on equation (4), the governing equations with respect to mechanical deformation and electrical potential turned into:

(5) 
where refers to the twodimensional Laplace operator and refers to the density of the piezoelectric material

(6) 

(7) 
The substitution

(8) 
reduces equation (5) into

(9) 
where


We consider the following solution of equation (9):

(10) 
where k and c are the wavenumber and phase velocity respectively, and are the respective solutions of

(11) 
where
and .
So the solutions of equation (11) are

(12) 
where are arbitrary constants.
Therefore the displacement components and the electric potential can be written as:

(13) 

(14) 
The constitutive equation for a fiberreinforced linearly elastic anisotropic medium is given by(Belfield et al. [36])
.
where are stress components, are the strain components, indicates the Kronecker delta.
refers to the preferred direction of reinforcement so that . The vector may be a function of position where the indices assume values from 1,2,3. Here , and represents reinforced parameters. identified as the shear modulus in transverse shear along the preferred direction, refers to the shear modulus in longitudinal shear in the preferred direction. , remains as specific stress components relevant to dissimilar layers for the concrete part of the composite material and is Lame’s constant.
Thus the equation of motion for the propagation of SH waves is represented as

(15) 
where
and are the rotational components in half space.
Therefore equation (15) becomes

(16) 
which becomes

(17) 
where
, 
, 

, 
, 

Assuming the solution of equation (17) in the form of

(18) 
where k and c indicate the wavenumber and wave velocity respectively.
Solving equation (17) becomes



(19) 
The solution of equation (19) yields the expression for the nonvanishing displacement component of halfspace as where
(C is an arbitrary constant)

(20) 
,

(21) 

(22) 
For electrically open case: electrical displacement turns into zero, i.e.,


(23) 
and
For electrically shorted case: electric potential turns to be zero, i.e.,


(24) 
Using the equations (13) and (14) and all the above boundary conditions, we get

(25) 

(26) 

(27) 

(28) 

(29) 

(30) 
where
Solving (25) to (29) we get





(31) 
whose real part is
And imaginary part is
Solving (25)(28) and (30) we get

(32) 







If we remove initial stresses from the piezoelectric layer and fiberreinforced half space i.e., and then the equation becomes

(33) 
where
and

(34) 







For a demonstration of the frequency equation of Love waves in a prestressed piezoelectric layer over a prestressed isotropic & homogeneous halfspace, we make . Then both equations become

(35) 
where ,
and

(36) 







For derivation of the frequency relation of the Love waves in an isotropic, homogeneous substrate embedded on an inhomogeneous semiinfinite space in the absence of prestress, and reducing the piezoelectric coefficient into zero, i.e., , , then the dispersion equation of (31) and (32) become in both cases as,

(37) 
where
and

(38) 







The dispersion equations of Love wave in prestressed isotropic, homogeneous layer over prestressed isotropic & homogeneous halfspace is obtained by considering , .

(39) 
where
and

(40) 



If we neglect the prestress in Case IV i.e., by setting , and , we get the dispersion equation as follows:

(41) 
where
and

(42) 
Therefore, for both cases, the dispersion equations (31) and (32) reduce to the following:
The above equation is the classical form of the Love Equation in a stressfree piezoelectric layer lying over a stressfree fiberreinforced halfspace.
For studying the impact of prestress on the propagation of Love waves in a piezoelectric substrate embedded over fiberreinforced halfspace, the values from equations (31) and (32) have been derived and graphically represented and illustrated. The figures represent the variation of nondimensional phase velocity with respect to the dimensionless wave number using the relevant values from Table 1 and Table 2.
Table 1. Values of Material coefficients of the piezoelectric material (PZT4) (Nie et al. [24])




25.6 
12.7 
6.45 
7500 
Table 2. The following data has been taken into consideration for the fiberreinforced half space under initial stress (Markham [25]):



7.07 
3.50 
1600 
For the numerical computation, we have considered the thickness (h) of the piezoelectric layer as 1.0, 1.25, 1.5, and 1.75 respectively. The relevant curves in the above figure 2 reflect that gradual elevation of the thickness of the piezoelectric layer tends to increase the phase velocity of the transmitting Love wave gradually which means that the thickness of the upper layer has a significant effect on Love wave phase velocity.
Fig. 2. Variation of Dimensionless phase velocity against dimensionless wavenumber for different values of depth
Fig. 3. Variation of Dimensionless phase velocity w.r.t dimensionless wavenumber with different values of and where
From this figure, we can say that with the increased value of and decrease the value of phase velocity also increases.
Fig. 4. Variation of Dimensionless phase velocity w.r.t dimensionless wavenumber with different values of and where both the prestresses P_{1} and P_{2 }à 0
Fig. 5. Variation of Dimensionless phase velocity w.r.t dimensionless wavenumber with different values of and where both and From this figure, we can say that with the increase value of , phase velocity also increases.
The curves in the above figure 3 show that the increasing values of the reinforcement parameter and accordingly decreasing values of the reinforcement parameter , resulted in an increase in the phase velocity of the Love waves. It is clear from this figure that an increase in the reinforcement parameter bounded with the expression favors the phase velocity of Love waves by inflating it.
Due to intrinsic lightweight properties and having high material strength and stiffness, fiberreinforced composites can be used in utilitarian aspects like industries relevant to construction, aviation, and medical instruments. The materials can significantly reduce damages in adjoining surfaces even due to severe vibrational impacts. In spite of having a wide range of applications in various engineering applications, piezoelectric materials carry drawbacks like reduced levels of piezoelectric constants, shape control, and high acoustic impedance which may result in lowering the control on derived distributed actuators. Keeping view of that to improve the efficiency levels, the proposed model of fiberreinforced piezoelectric composite materials with the presence of prestress has been studied for both electrically open and shorted cases and thereby the dispersion relations have been established with the aid of variableseparable method by which we can easily eliminate the firstorder differential terms for further simplification. From the dispersion relation and subsequent graphs, the outcomes can be briefed as:
In view of the consequences drawn from the study, the results are useful for designing highmode Love mode sensors and can also be used for developing medical imaging instruments and underwater transducers. Precisely, the present study is likely to prove its efficiency in modeling the damages caused by the artificial explosion of the layered piezoelectric reinforced composite structures. Since the properties of SAW devices specifically depend on phase delay, therefore, from a utilitarian aspect the outcomes can be employed in the designing of Love wave sensors.
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
The author declares that there is no conflict of interest regarding the publication of this article.