The present work investigates the mechanically forced vibration of the hydro-elasto-piezoelectric system consisting of a two-layer plate “elastic+PZT”, a compressible viscous fluid, and a rigid wall. It is assumed that the PZT (piezoelectric) layer of the plate is in contact with the fluid and time-harmonic linear forces act on the free surface of the elastic-metallic layer. This study is valuable because it considers for the first time the mechanical vibration of the metal+piezoelectric bilayer plate in contact with a fluid. It is also the first time that the influence of the volumetric concentration of the constituents on the vibration of the hydro-elasto-piezoelectric system is studied. Another value of the present work is the use of the exact equations and relations of elasto-electrodynamics for elastic and piezoelectric materials to describe the motion of the plate layers within the framework of the piecewise homogeneous body model and the use of the linearized Navier-Stokes equations to describe the flow of the compressible viscous fluid. The plane-strain state in the plate and the plane flow in the fluid take place. For the solution of the corresponding boundary-value problem, the Fourier transform is used with respect to the spatial coordinate on the axis along the laying direction of the plate. The analytical expressions of the Fourier transform of all the sought values of each component of the system are determined. The origins of the searched values are determined numerically, after which numerical results on the stress on the fluid and plate interface planes are presented and discussed. These results are obtained for the case where PZT-2 is chosen as the piezoelectric material, steel and aluminum as the elastic metal materials, and Glycerin as the fluid. Analysis of these results allows conclusions to be drawn about the character of the problem parameters on the frequency response of the interfacial stress. In particular, it was found that after a certain value of the vibration frequency, the presence of the metal layer in the two-layer plate led to an increase in the absolute values of the above interfacial stress.

In the classical sense, the study of the dynamics of a piecewise homogeneous acoustic medium, such as “plate+fluid” systems in the cases where the plate is made of conventional metal or polymer materials, is necessary for various fields of modern industry, e.g., fluid transport, geophysical investigations, biomedical device construction, submarine construction and sound insulation. The results of these studies provide a theoretical basis for understanding the acoustic phenomena occurring in the piecewise homogeneous acoustic medium. Consequently, these results provide opportunities for the control and management of these phenomena.

In modern literature, the work of Lamb [

Note that these investigations can be classified according to various aspects, such as the theories applied to describe the motion of the plate (approximate plate theories, which are described, for example in [

The aforementioned works serve not only their direct purpose but also provide the theoretical basis for the study of the corresponding problems related to the cases in which the plate material is intelligent, such as piezoelectric or piezomagnetic. It should be noted that the results of such investigations have great importance for energy harvesting procedures (see, for example, references [

Nevertheless, there has not been enough fundamental theoretical research in this field, to which the present research is dedicated, in particular, to the mechanical forced vibration of the hydro-piezoelectric system consisting of a two-layer “metal+PZT” plate, a compressible viscous fluid, and a rigid wall. The research presented here is carried out in the framework of the piecewise-homogeneous body model using the exact equations and relations of elastodynamics and electrodynamics in the plane-strain state to describe the motion of the “metal+piezoelectric” plate and using the linearized Navier-Stokes equations for the plane flow of the compressible viscous fluid.

To show the relevance and importance of the present work, let us consider a brief review of related research and begin with the work of Belkourchia et al. [

The piezoelectric composite cantilever beam model is also used in the work of He et al. [

In contrast to the aforementioned work, the work by Amini et al. [

The work of Nie et al. [

The paper by Qiao et al. [

In all the above works dealing with hydro-piezoelectric systems, the fluid flow is described in the framework of the incompressible viscous or inviscid Newtonian fluid model using the linear Navier-Stokes equations. Moreover, in these works, the motion (or vibration) of the piezoelectric cantilevers is considered with finite sizes, so such a model cannot discover the local electromechanical and hydro-electromechanical coupling effects without the influence of the boundary conditions. Such an effect related to the subject of the present work can be discovered in the framework of infinite two-layer “metal+piezoelectric” plate and fluid systems.

Note also that the above work does not investigate how the electromechanical coupling effect of the piezoelectric material can affect the pressure at the interface between the fluid and the plate. Since the energy generation in the energy harvesting piezoelectric systems comes from this pressure, the above coupling effects must be considered in theoretical studies.

In order to address the above issues, Ekicioglu Kuzeci [

In the present paper, an attempt is made to develop these investigations for the system consisting of a two-layer “metal+piezoelectric” plate, a compressible viscous fluid, and a rigid wall. The motion of the plate is described within the piecewise-homogeneous body model using the exact equations of elasto-and electrodynamics for piezoelectric materials, and the flow of the fluid is described using the linearized Navier-Stokes equations for a compressible viscous fluid.

The present work can also be considered as further development of the second author’s work from the last decade dealing with the forced vibration of single-layer purely elastic or viscoelastic plates in contact with a compressible viscous fluid (see, for example, the works [

Thus, it is clear that the investigations on the subject of the present work were carried out for the case where the plate is single-layered. Thus, the main difference of the present work from the previous related works of the second author and his collaborators, as well as from the investigations described in the works [

Note that in the formulation of the problem studied in this paper, we use the linearized Navier-Stokes equations obtained from the corresponding nonlinear Navier-Stokes equations for compressible barotropic viscous fluids. Linearization is one of the methods for solving nonlinear problems. Many other methods for solving nonlinear physical-mechanical problems have been developed in recent years. We briefly review some here and begin with the work of Mahdy et al. [

Moreover, in a series of studies by Bazighifan et al. [

In this context, since the present study refers to coupling field problems, we refer to the recent studies of Mahdy et al. [

The remainder of this paper is organized as follows. The mathematical formulation of the problem, the equation of motion of the metal/PZT plate and the equations of flow, as well as the corresponding boundary, contact, and compatibility conditions are given in Section 2. The solution method for the formulated boundary value problem and the algorithm for obtaining the numerical results are explained in Section 3. The numerical results are presented and discussed in Section 4, and finally, the results are summarized in Section 5.

We consider the hydro-elastic-piezoelectric system shown in

In this framework, it is required to determine the mechanical and electrical fields in the above hydro-elastic-piezoelectric system by using the appropriate exact field equations and relations. For this purpose, we write these field equations and relations for each component of the system, using the upper indices (m) and (p) to indicate the affiliation of the quantities to the metallic and piezoelectric layers, respectively. According to the monograph by Yang [

In addition, the following relations are:

Mechanical relations for the metal layer material:

Electro–mechanical relations for the piezoelectric layer material when it is polarized along the

In

In

This completes the field equations and relations describing the motion of the two-layer “metal+piezoelectric” plate. Now we try to write the field equations and relations for the compressible (barotropic) viscous fluid flow. According to the monograph of Guz [

In

According to the monograph by Guz [

Now we try to formulate the corresponding boundary conditions at the

The boundary conditions on the upper face plane of the plate are:

Perfect contact conditions on the plane between the metal and piezoelectric layers are:

Compatibility conditions on the interface plane between the fluid and plate are:

The impermeability conditions on the rigid wall are:

We also add the boundary conditions on the planes

This completes the formulation of the problems investigated in the present paper.

The presence of piezoelectric components in composite materials complicates the analytical solution of the corresponding problems. Therefore, it sometimes seems necessary to apply numerical methods, as in the works of Aylikci et al. [

According to the definition of the time-harmonic oscillation, let us represent all the sought values of the considered problem as

According to which, the amplitudes can be presented as follows:

Now, we consider separately the determination of the Fourier transforms of the quantities related to the metal and piezoelectric layers of the plate and related to the fluid.

Substituting the expressions related to the metal layer given in

Using the relations

Substituting the expressions in

We recall that in

In

Thus, substituting the particular solutions in

According to the requirement of the existence of the non-trivial solution of the system of equations in

In

We employ Vieta’s trigonometric formula for cubic equations in order to find the roots of the

In the next step, the values of the expression in

According to the sign of

However, if

In this way, we determine the roots of the characteristic

According to the expressions in

Substituting the expressions in

Thus, the expressions of the Fourier transforms of the values related to the piezoelectric plate are completely determined.

To determine the Fourier transforms of the quantities related to the fluid flow, first we consider determination of

Substituting the expressions in

Note that the dimensionless numbers

Thus, according to the well-known solution technique of ordinary differential equations, the solution to the equations in

Using the expressions in

Finally, we obtain a system of 14 linear algebraic equations which contains 14 unknown constants by substituting the expressions in

Originals of the sought values, i.e., the integrals in

As in the case under consideration, the fluid is modelled as a viscous one, therefore, the integrals in

Note that under the calculation procedure, the improper integrals

In the numerical investigations, steel and aluminum are selected as the metallic materials and PZT-2 as the piezoelectric material. According to Yang [

Properties of materials | Materials | ||
---|---|---|---|

Aluminum | Steel | PZT-2 | |

2.60 | 7.75 | 2.22 | |

10.20 | 24.76 | 13.5 | |

5.0 | 9.26 | 6.81 | |

10.20 | 24.76 | 11.3 | |

2700 | 7795 | 7600 | |

- | - | −1.9 | |

- | - | 9.0 | |

- | - | 9.8 | |

- | - | 8.7615 | |

- | - | 3.9825 | |

5.0 | 9.26 | - | |

2.60 | 7.75 | - |

We consider the numerical results related to the frequency response of the dimensionless normal stress

To begin, we consider the frequency response of the dimensionless interface stress

Let us now consider the graphs illustrated in

From analysis of the results shown in

The graphs of the frequency response of the interfacial stress have a smooth character, and the absolute values of this stress increase monotonically with the vibration frequency.

An increase in the thickness of the plate leads to an increase in the absolute values of the stress, and the character of the frequency responses constructed for different plate thicknesses is the same.

An increase in the values of the ratio

From analysis of the results illustrated by the graphs in

The character of the interface stress frequency response graphs depends significantly not only on the plate thickness, but also on the fluid depth, i.e., on the ratio

For relatively large values of plate thickness, there is a low-frequency region of frequency change (denote this diapason as

There is such a value of frequency

Increasing the thickness of the plate generally leads to an increase in the absolute values of the stress.

For a relatively large plate thickness, a resonant frequency occurs for the considered hydro-elastic system and the values of the resonant frequencies increase with the ratio as well as with the plate thickness.

The character of the influence of the ratio

Comparison of the results for the metal plate case with the corresponding results for the piezoelectric plate shows that the absolute values for the metal plate are significantly higher than the corresponding values for the piezoelectric plate.

Consequently, piezoelectricity significantly complicates the frequency responses of the interfacial stress, not only in a quantitative sense, but also in a qualitative sense.

Now the question arises of how the mixture of metal and piezoelectricity affects the above frequency response. To answer this question, we consider and analyze the results presented in

We introduce the notation

At the same time, comparison of the results in

Recall that all the above results hold for the case where

These results are obtained for the cases with

Moreover, it is evident from these results that before (after) a certain value of this frequency, an increase in the values of the ratio

We have analyzed the results of the two-layer “steel+PZT-2” plate and all previous conclusions refer to this case. Now we consider some results shown in

Thus, it follows from the results in

Finally, we consider the numerical convergence of the used computational algorithm within the framework of which the above results are obtained. Note that the convergence of the numerical results with respect to the length of the integration interval, i.e., with respect to

Thus, in the present work, the mechanically forced vibration of the hydro-elastic-piezoelectric system consisting of a two-layer “elastic+PZT” plate, a compressible viscous fluid, and a rigid wall has been investigated. The structure of this work is organized as follows: in Section 1, an overview and the findings of related research are given. In Section 2, the mathematical formulation of the problem, the equation of motion of the metal/PZT plate and the flow equations, as well as the corresponding boundary, contact, and compatibility conditions are given. In Section 3, the solution method for the formulated boundary value problem and the algorithm for obtaining numerical results are explained. The numerical results are presented and discussed in Section 4 and, finally, the results are summarized in the present section, i.e., in Section 5.

The PZT layer of the plate is assumed to be in contact with the fluid and time-harmonic linear forces act on the free surface of the elastic-metallic layer. The motion of the plate is described within the piecewise homogeneous body model using the exact equations and relations of elastodynamics and elasto-electrodynamics for piezoelectric materials. The fluid flow is described by the linearized Navier-Stokes equations for the barotropic, compressible, viscous Newtonian fluid. The plane-strain state in the plate and the plane flow in the fluid take place. For the solution of the corresponding boundary-value problem, the Fourier transform is used with respect to the spatial coordinate on the axis along the laying direction of the plate. The analytical expressions of the Fourier transform of all the sought values of each component of the system are determined. The origins of the searched values are determined numerically, after which the numerical results on the stress on the fluid and plate interface planes are presented and discussed.

These results are obtained for the case where PZT-2 is chosen as the piezoelectric material, steel and aluminum as the elastic metal materials, and glycerin as the fluid.

Analysis of these results allows the following concrete conclusions to be drawn about the character of the influence of the problem parameters on the frequency response of the stress acting on the interface between the piezoelectric layer and fluid:

The character of the interface stress frequency response graphs obtained for the two-layer “metal+piezoelectric” plate depends significantly not only on the total thickness

At low vibration frequencies of the two-layer “metal+piezoelectric” plate and the plate consisting only of piezoelectric material (i.e., the “piezoelectric” plate), the zone “descent+relatively sharp rise+beginning of the last descent” (D + RSR + BLD) appears.

Assuming that the D + RSR + BLD zone for the two-layer “metal+piezoelectric” plate appears for the frequencies

As in the case where the plate is made of piezoelectric material only, and as in the case of the two-layer “metal+piezoelectric” plate, the resonance case occurs and the resonance frequency obtained for the two-layer “metal+piezoelectric” plate is lower than that obtained for the “piezoelectric” plate.

The resonance frequency of the two-layer “metal+piezoelectric” plate decreases with the thickness of the metal layer in the plate.

It is found that

The differences

The mechanical properties of the metal layer in the plate have a significant influence on the frequency response of the stress under investigation.

Unfortunately, we did not find any related results from other authors with which to compare the present results. Therefore, we compared the obtained results with those of the authors of the present work for the cases where the materials of the plate layers are the same, i.e., the same metal material or the same PZT material.

The above comparison shows that the concrete numerical results obtained are in agreement with known physical-mechanical considerations, and made it possible to control the magnitude of the interfacial pressure between the fluid and the plate by the choice of materials and the thickness of the plate layers.

In future works, an electrical field instead of mechanical force can be applied to the piezoelectric plate, so the mechanical response of the system can be analyzed. Furthermore, interface stress can be studied for different PZT and metal combinations.