Introduction. To improve the efficiency of piezoelectric transducers, research is being conducted to find the optimal configuration of the device — its size, shape, electroplating pattern, loading methods, and the materials used. Despite a significant amount of work having been carried out, the task of determining efficient designs for piezoelectric generators and selecting piezo-materials with improved characteristics for energy storage devices remains a challenge. One of the most promising ways to improve the performance of such transducers is the use of piezocomposite materials. The most well-known and thoroughly studied are fibrous piezocomposites, or systems with a 1–3 connectivity pattern according to Newnham's terminology [1], in which rod-shaped piezoceramic elements are embedded in an elastic dielectric matrix. Among numerous papers devoted to 1–3 composites, we should note [2], which examines applications for hydrophones. In [3], the efficient properties are calculated through solving boundary value problems by the finite element method for a unit cell with periodic boundary conditions. In [4], the corresponding coefficients are presented in closed form. In [5], formulas are derived based on asymptotic averaging, and comparison with experiment shows good agreement. Homogenization theory for the design of optimal piezocomposites is applied in [6]. In [7], analytical expressions for the effective constants are also obtained under the condition that the structure period is significantly smaller than the elastic wavelength, and new composites with improved global properties for biomedical imaging are proposed. An analysis of electroelastic composites using self-consistent methods and asymptotic homogenization is performed in [8]. In [9], a theoretical framework for designing piezocomposites with specified generalized characteristics is presented. Additional information is gathered in review [10]. 1–3 composites provide high values for a number of key parameters important for applications — the hydrostatic piezoelectric charge coefficient, the hydrostatic voltage coefficient, the thickness electromechanical coupling coefficient, the hydrostatic Q-factor, and others.

Macro fiber composites, originally developed for the aerospace industry, are also successfully used in energy harvesting systems. These miniature, low-power generators are thin beams and include piezoelectric fibers placed in a dielectric medium. The fibers are integrated into a single multilayer structure with interdigital electrodes on the lateral surfaces of the beams on both sides. The objective of [11] was to assess the effect of the uncertainty in the physical properties of both the piezoelectric fibers and the epoxy matrix on the model response, and to identify the parameters that determine the greatest variability in the output data. In [12], a potential technology is proposed for integrating energy harvesting sources into complex airframe structures in aerospace engineering, enabling the generation of the energy required for monitoring environmental conditions or controlled structural characteristics. A comparative analysis of generators made from monolithic and composite piezoceramics, conducted in [13], showed the high efficiency of using macro fiber composites. Study [14] presents the development of a microgenerator based on microcomposite piezoelectric materials for energy harvesting in glove structures. The devices described consist of piezoelectric fibers with a diameter of 90–250 μm, aligned in a unidirectional order and embedded in a composite structure.

To improve the performance of 1–3 composites, it is possible to vary the materials of the piezoelectric fibers and the elastic dielectric matrix. In particular, porous materials with a reduced Young's modulus and a significantly varying Poisson's ratio can be used as the matrix [2][14]. In [15][16], the Mori-Tanaka method is used to determine the effective moduli first for the porous matrix, after which an analysis of 1–3 composites is performed. In [15][16], additional options for pore orientation relative to the polarization direction are considered, which, however, is difficult to implement in practice. In [2], approximations of the effective environmental method are used. In [17][18] a topological optimization of the porous structure of the matrix is performed based on the approaches from [2]. In [19], a modeling of a 0–3 composite with piezoelectric particles in a porous matrix is performed. It is also shown that the use of a porous matrix does not lead to an improvement in the electromechanical properties of a 0–3 system. However, the piezoelectric sensitivity coefficients are not analyzed in that study.

Another direction of modification is related to the use of porous piezoceramics, which in recent years have been considered a promising active material for energy storage devices. Compared with dense ceramics, porous materials are characterized by reduced acoustic impedance, increased piezoelectric sensitivity, and a number of high-quality indicators. Thus, in [20], a Rosen-type piezoelectric element utilizing the longitudinal piezoelectric modulus is investigated. In [21], it is shown that introducing a porous layer into a multilayer element based on barium titanate significantly improves performance in piezoelectric energy harvesting. Experimental study [22] describes a piezoelectric generator manufactured using porous ceramics. In [23], a review of the current state and prospects for the use of porous piezoceramics is given.

The present study builds upon the results of [20][24] and, to a certain extent, combines approaches to the introduction of porous piezoelectric ceramics as an active phase in 1–3 composite fibers and to varying matrix materials with different stiffness. In other words, it examines 1–3 piezoelectric composites in which porous piezoceramics are used as the piezoelectrically active material.

Thus, an analysis of the literature shows that the main efforts of researchers are focused on determining the efficient electroelastic properties of 1–3 composites. A wide range of methods has been developed — from analytical formulas to detailed numerical procedures — that make it possible to predict with high accuracy the homogenized (averaged) characteristics of the material depending on the properties of the constituent phases: the fibers and the matrix. This information is undoubtedly important, but it describes the behavior of the material as a whole, rather than that of the final structural element, which is the goal of engineering design.

The transition from efficient properties to the analysis of the dynamics of real structures, such as plates and shells, represents an independent and non-trivial problem. Although the finite element method (FEM) is applicable for high-precision analysis, as implemented in numerous software packages, full-scale three-dimensional modeling at the stages of preliminary design and optimization often proves to be excessively resource-intensive. At the same time, the development of applied two-dimensional theories (theories of plates and shells) that take into account the specific features of 1–3 composites (particularly in the presence of porous 3–0 and 3–3 components) would significantly simplify and speed up calculations, while maintaining accuracy sufficient for engineering applications. In the scientific literature, there is a distinct lack of such models capable of acting as a bridge between the micromechanics of the material and the macromechanics of the structure.

The objective of this work is to construct an applied two-dimensional theory for calculating the transverse vibrations of layered plates made of 1–3 piezoelectric composite, followed by validation of the results obtained using the proposed theory through comparing them with a solution from the ACELAN finite element package.

Materials and Methods. The development of a mathematical model based on the linear theory of electroelasticity and a computer model of a multilayer transducer with polarization along the layers is performed in the ACELAN finite element package. The computer model of the device is subsequently used to validate the adequacy of the developed applied theory.

Mathematical model

The paper considers models of piezoelectric transducers consisting of electroelastic composite materials, whose properties are specified by effective constants [20][24]. Let us introduce index j for numbering the bodies and write down the equations and constitutive relations [25]:

(1)

where σ — stress tensor; ρj — density of the body; ε — strain tensor; u — displacement vector; D — electric induction vector; E— electric field strength vector; fj — vector of mass forces; φ — electric potential; α, β, ς, βdj, ςd — damping coefficients;

, ,  — tensors of elastic constants, piezo moduli and permittivity; index j corresponds to the number of the body in the model.

System (1) is supplemented with the corresponding mechanical and electrical boundary conditions. In particular, the value of the electric potential φ is specified on the electrodes, and on the non-electrode part of the surface — the no-charge condition is set, i.e., the normal component of the electric induction vector D is equal to zero. In the case of free electrode SE, potential V0 on it is unknown and is determined from an additional condition:

(2)

The same condition is applied when determining the antiresonance frequencies.

To analyze the efficiency of the device, the values of the output electric potential, mechanical stresses, and the electromechanical coupling coefficient (EMCC) are used. This is calculated from the formula:

(3)

where fr and fa — resonance and antiresonance frequencies, respectively.

The calculation of the damping coefficients when determining the output potential at resonant frequencies was performed using the approach from [25], for which a constant quality factor Q was assumed at the first two resonant frequencies:

(4)

The operating frequencies (fr1) and the next closest ones (fr2) are selected as such resonant frequencies. For the structures under consideration, the operating frequency is the first bending mode.

When modeling porous composites, the efficient moduli obtained for porous PZT-4 piezoceramics by the averaging method [20][26] in the ACELAN-COMPOS package were used. Data were obtained for materials with porosity up to 80%. The efficient moduli used in the numerical experiments are given in Table 1.

When modeling a 1–3 composite (Fig. 1), the effective properties (Tables 2, 3) are taken from [24].

Layered plate structures and finite element model

The piezoelectric element under consideration is a layered plate (Fig. 2), in which each layer is a 1–3 piezocomposite, with the piezoceramic rods (marked by black circles in Fig. 2) made of porous piezoceramics. It is noteworthy that two layers symmetric with respect to the midsurface have opposing longitudinal polarization.

In the case where the piezoelectric element consists of two layers, its finite element model is constructed. Figure 3 a presents the geometric model with two oppositely polarized layers (red and yellow colors), and Figure 3 b shows the mesh of triangular quadratic finite elements (1350 finite elements, 2839 nodes).

The finite element model constructed in this section was used to validate the adequacy of the applied theory in calculations in the following sections.

Research Results

Development of an applied theory for calculating transverse vibrations of a layered structure with polarization of layers in the plane of the plate

The applied theory of bending vibrations of the plate under consideration, consisting of four layers, in which the upper and lower layers are isotropic (thicknesses H with Lamé coefficients λ, μ), and the internal electroelastic thickness h is constructed on the basis of Kirchhoff-Love-type hypotheses. The first hypothesis is that normal stresses are equal to zero throughout the entire volume of the plate σ11 = 0, which makes it possible to express longitudinal deformations:

(5)

The second hypothesis shows that the distribution of the components of the displacement vector and the electric potential has the form:

(6)

where UX(y, z) — deflection of the midsurface, x1 = x, x2 = y, x3 = z.

Taking into account relations (5) and (6) in system (1) in the case of steady-state vibrations with circular frequency ω leads to a system of equations for two unknown functions Φ(y, z), UX(y, z).

(7)

Bending moments and shear forces have the form:

For the case of a panel consisting of two internal layers (Fig. 2 and 3 a), the second equation of system (7) is reduced to the form:

(8)

from which the electric potential is expressed:

(9)

After this, the first equation (7) is reduced to the form:

(10)

System of equations (7) is generally solved numerically, whereas equation (10) can be solved analytically.

Validation of the proposed method for adequacy based on a comparison of the results with calculations in the ACELAN package

To test the applicability of the proposed applied theory, two problems for a two-layer panel are solved: determining the first resonant frequency of the bending mode, and the problem of steady-state vibrations at a frequency of 200 Hz under a uniformly distributed load on the upper plane of the plate with an amplitude of 1000 Pa. The plate length is 0.1 m, the thickness of each layer is 0.0025 m, and the material is PZT-4. The plate is fixed at the left end and hinged at the right (Fig. 3 a).

The self-resonant frequency of the first bending mode (Fig. 4) in the ACELAN package calculation is 1348.68 Hz, while the applied theory calculation is 1360 Hz, i.e., the error is 0.8%.

The forced vibration calculation has yielded the following results: Figure 4 shows the distribution of vertical displacements on the deformed plate.

Figure 5 shows the graphs of the deflection of the midsurface of the plate.

Figures 6 and 7 present similar results for horizontal displacement, and Figures 8 and 9 show them for electric potential.

Discussion. The key result of this research is an applied two-dimensional theory and the demonstration of a high degree of agreement between the solutions obtained from it and the simulation data in the ACELAN finite element package. The deviation in the calculation of the first resonant frequency is only 0.8%, which is more than sufficient for engineering applications. Visual and quantitative comparison of the graphs of deflections, horizontal displacements, and the distribution of electric potential under forced vibrations also shows their almost complete agreement (Figs. 5, 7, 9), which confirms the proposed model adequacy in the frequency range under consideration.

The primary application of the developed theory is the low-frequency range. The structural geometry is thin plates. The range of variation of the material parameters of the layers is relatively small. For wider property ranges, when constructing the applied theory, it will be necessary to take into account the broken normal and use a higher order of approximation for the characteristics of the mechanical and electrical fields across the thickness of the structure.

The proposed theory eliminates resource-intensive 3D modeling at the preliminary design stages. The practical significance of the results is in the creation of an efficient and fast tool for engineers and researchers: instead of time-consuming calculations in “heavy” CAE packages, analytical or numerically analytical solutions of equations (7)–(10) can be used to quickly evaluate dynamic characteristics, conduct parametric studies, and optimize the designs of piezoelectric transducers, sensors, and energy harvesting devices.

Conclusion. Within the framework of the study, the problem of constructing and validating an applied two-dimensional theory for calculating the transverse vibrations of layered plates made of a 1–3 piezocomposite and porous piezoactive 3–0 and 3–3 elements is successfully solved.

A mathematical model based on the Kirchhoff-Love hypothesis is developed to describe the dynamic behavior of a thin layered electroelastic plate with opposing longitudinal polarization of the layers.

A system of differential equations is derived, which, for the particular case of a two-layer panel, reduces to a single equation that admits an analytical solution.

The proposed theory is validated through comparing its results with numerical simulation data in the ACELAN finite element package. The error in determining the first resonant frequency does not exceed 0.8%.

The adequacy of the theory in calculating forced vibrations is proven: the profiles of vertical and horizontal displacements, as well as the distribution of electric potential, calculated using the applied theory, almost completely coincides with the results of finite element modeling.

Thus, it is shown that the proposed two-dimensional theory is an effective and computationally efficient tool for the analysis and design of thin-walled structures made of piezocomposite materials, which allows for a significant acceleration of development compared with the use of universal finite element packages.

Prospects for further research are related to the development of the next level of homogenization — the transition to the efficient properties of a 2–2 composite with subsequent consideration of a two-layer plate — as well as with more complex approximations of mechanical and electrical fields, including consideration of edge effects. However, when complicating the model, it should be remembered that the primary practical purpose of such theories is the calculation of the output integral characteristics of devices: resonant frequencies, output electrical potential, power, etc.