Multilayered Piezoelectric Plates Research Articles

Tunable bandgaps of guided waves by periodic shunting circuits in multilayered piezoelectric plates

The multi-mode and dispersion characteristics of guided waves in plate bring great challenges to the manipulation of guided waves and controlling vibration and noise. In this study, the Stroh formalism is developed to establish a theoretical model for deriving the dispersion relations of guided waves in a multilayered piezoelectric plate. The dispersion characteristics under different electrical boundary conditions introduced by periodic shunting circuits are discussed to reveal the influences of external circuits on guided wave propagation. The proper shunting circuit is not only the key point to obtain the specific frequency of bandgap, but also the effective method to control the bandwidth. Furthermore, for the separation of symmetric and anti-symmetric modes of guided waves, the multilayered piezoelectric plate with a mirror plane is studied by using an equivalent model. Results indicate that the propagation behaviors of guided waves in multilayered piezoelectric plates can be interpreted by developed Stroh formalism, and their bandgaps can be effectively manipulated by periodically external circuits. The tunable bandgaps in this work could be helpful for designing smart devices of wave isolation based on multilayered piezoelectric plate.

Electromechanical coupling characteristics of multilayered piezoelectric quasicrystal plates in an elastic medium

AbstractQuasicrystals (QCs) have attracted tremendous attention of researchers for their unusual properties. In this paper, an exact electric‐elastic solution of the simply supported and multilayered three‐dimensional (3D) cubic piezoelectric quasicrystal (PQC) nanoplate with the nonlocal effect is derived. Based on the basic elasticity equation of 3D QCs, we construct the linear eigenvalue system in terms of the pseudo‐Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. The two‐parameter foundation model is utilized to simulate the interaction between the nanoplate and elastic medium. The propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. Based on the boundary conditions of the upper and lower surfaces of the laminate and foundation model, the solutions are employed to derive from the global propagator matrix. Compared with the conventional propagator matrix method, a new propagator method is reestablished to deal with numerical instabilities of the case of large aspect ratio and high‐order frequencies for QC laminates. Finally, typical numerical examples are presented to illustrate the influence of nonlocal parameters and elastic medium coefficients on phonon, phason, and electric variables of 3D PQC nanoplates.

Static solution of two-dimensional decagonal piezoelectric quasicrystal laminates with mixed boundary conditions

Piezoelectric quasicrystals have attracted the tremendous attention of researchers for their unique properties. In this paper, the semi-analysis solutions of the functionally graded multilayered two-dimensional decagonal piezoelectric quasicrystal plates are investigated for mixed boundary-value problems. Based on the quasicrystal linear elastic theory, the state-space method is employed to derive the state equations composed of partial differential along the thickness direction. The differential quadrature method is utilized to discretize state variables to satisfy the mixed boundary conditions in the horizontal plane. Different from the conventional propagator matrix, a new propagator relationship is proposed to study numerical instability caused by extensive discrete points. Comprehensive numerical results are shown for the sandwich quasicrystal plate with four different stacking sequences. The numerical results reveal the effect of the functional gradient index factors, and boundary conditions on the electric potential, electric displacements, stresses, and displacements under the mechanical/electric loadings.

Guided wave propagation in multilayered periodic piezoelectric plate with a mirror plane

The multi-mode and dispersion natures of guided waves in plate bring big challenges of functional phononic crystal (PC) plate design for wave control. For designing a periodic multilayer piezoelectric plate to control specific guided wave propagation, it is vital to study the dispersion relation of each guided wave mode. In this paper, a plane wave expansion method is developed to calculate the dispersion relation of each guided wave mode in the periodic multilayer piezoelectric plate with a mirror plane. To quantitatively demonstrate the effectivity of the developed method, the band structures of PC plates with different kinds of piezoelectric scatterers are discussed to reveal the influences of geometry and polarization effect of piezoelectric scatterer on the guided wave propagation. Results show that the mode bandgaps of symmetric mode Lamb waves, anti-symmetric mode Lamb waves, and shear horizontal waves can be obtained separately by the proposed method, and they exhibit different responses with the changes of piezoelectric scatterer. Therefore, the proposed method can provide an effective analysis way of band structures for designing smart guided wave control devices based on multilayered piezoelectric PC plate.

Layer-Wise Discontinuous Galerkin Methods for Piezoelectric Laminates

In this work, a novel high-order formulation for multilayered piezoelectric plates based on the combination of variable-order interior penalty discontinuous Galerkin methods and general layer-wise plate theories is presented, implemented and tested. The key feature of the formulation is the possibility to tune the order of the basis functions in both the in-plane approximation and the through-the-thickness expansion of the primary variables, namely displacements and electric potential. The results obtained from the application to the considered test cases show accuracy and robustness, thus confirming the developed technique as a supplementary computational tool for the analysis and design of smart laminated devices.

SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces

In this paper, analytical solutions for SH waves in transversely isotropic multilayered piezoelectric semiconductor (PSC) plates with imperfect interfaces are obtained. The extended displacements and stresses are expressed in terms of the eigenvalues and eigenvectors by introducing the extended Stroh formalism. Making use of the dual variable and position (DVP) method and the imperfect interface conditions, the transfer matrix which relates the extended displacement and traction on the lower and upper interfaces of the multilayered plates is derived. Then the dispersion relation is obtained by using the boundary conditions on the top and bottom surfaces of the multilayered plates. Effect of the steady-state carrier density in single-layer ZnO plate, and effect of stacking sequences and imperfect interfaces in sandwich plates are discussed via numerical examples. Particularly, the critical elastic (E), piezoelectric (PE), and PSC wave domains are identified for the given carrier density, plate thickness and frequency, which could be very helpful as theoretical guidance for the design of PSC devices.

Static response of functionally graded multilayered one-dimensional hexagonal piezoelectric quasicrystal plates using the state vector approach

The effect of the non-homogeneity of material properties has been considered the important variation mechanism in the static responses of quasicrystal structures, but the existing theoretical model for it is unable to simulate the material change format beyond the exponential function. In this paper, we create a new model of functionally graded multilayered 1D piezoelectric quasicrystal plates using the state vector approach, in which varying functionally graded electro-elastic properties can be extended from exponential to linear and higher order in the thickness direction. Based on the state equations, an analytical solution for a single plate has been derived, and the result for the corresponding multilayered case is obtained utilizing the propagator matrix method. The present study shows, in particular, that coefficient orders of two varying functions (the power function and the exponential function) of the material gradient provide the ability to tailor the mechanical behaviors in the system’s phonon, phason, and electric fields. Moreover, the insensitive points of phonon stress and electric potential under functionally graded effects in the quasicrystal layer are observed. In addition, the influences of stacking sequences and discontinuity of horizontal stress are explored in the simulation by the new model. The results are very useful for the design and understanding of the characterization of functionally graded piezoelectric quasicrystal materials in their applications to multilayered systems.

Three-dimensional analysis of freely vibrating multilayered piezoelectric plates through adaptive global piecewise-smooth functions

In this article, freely vibrating multilayered piezoelectric plates are analyzed through a set of adaptive global piecewise-smooth functions along with governing differential equations and associated boundary conditions, which are consistently derived from the classical theorem of virtual displacements. The analysis demonstrates the capability of the adaptive global piecewise-smooth functions to treat any multilayered plate as if it were made up of a single layer even in the presence of multiphysics analyses such as piezoelectric layers. The relevant model is essentially two-dimensional because it is based on an expansion through the thickness of the plate aimed at modeling a three-dimensional dynamical behavior. In order to demonstrate the effectiveness of the model, all the results are compared to exact three-dimensional results; these latter are extracted through a three-dimensional model based on a transfer matrix technique whose numerical stability is achieved using scaled electric potentials. The exact graphical results are herein illustrated, thus showing both the effectiveness of using weighted electric potentials and the capability of adaptive global piecewise-smooth functions to converge at exact results through a minimum computational effort.

Quasi-3D Stability and Vibration Analyses of Sandwich Piezoelectric Plates with an Embedded CNT-Reinforced Composite Core

Quasi-three-dimensional (3D) stability and free vibration analyses of bi-axially loaded, simply-supported, sandwich piezoelectric plates with an embedded either a functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) core or a multilayered fiber-reinforced composite (FRC) one are presented. Three different distributions of carbon nanotubes (CNTs) through the thickness of the CNTRC core, i.e. uniformly distributed and FG V-, rhombus- and X-type variations, are considered, and the effective material properties of the CNTRC core are estimated using the rule of mixtures. The Pagano method, which is conventionally used for the analysis of multilayered FRC plates, is modified to be feasible for the study of sandwich hybrid CNTRC and piezoelectric ones, in which Reissner mixed variational theorem, the successive approximation and transfer matrix methods, and the transformed real-valued solutions of the system equations are used. The modified Pagano solutions for the stability and free vibration of multilayered hybrid FRC and piezoelectric plates are in excellent agreement with the exact 3D ones available in the literature, and those for sandwich hybrid CNTRC and piezoelectric plates may be used as the benchmark solutions to assess the ones obtained by using various 2D theories and numerical models.

Layerwise mixed models for analysis of multilayered piezoelectric composite plates using least-squares formulation

This work provides an assessment of layerwise mixed models using least-squares formulation for the coupled electromechanical static analysis of multilayered plates. In agreement with three-dimensional (3D) exact solutions, due to compatibility and equilibrium conditions at the layers interfaces, certain mechanical and electrical variables must fulfill interlaminar C0 continuity, namely: displacements, in-plane strains, transverse stresses, electric potential, in-plane electric field components and transverse electric displacement (if no potential is imposed between layers). Hence, two layerwise mixed least-squares models are here investigated, with two different sets of chosen independent variables: Model A, developed earlier, fulfills a priori the interlaminar C0 continuity of all those aforementioned variables, taken as independent variables; Model B, here newly developed, rather reduces the number of independent variables, but also fulfills a priori the interlaminar C0 continuity of displacements, transverse stresses, electric potential and transverse electric displacement, taken as independent variables. The predictive capabilities of both models are assessed by comparison with 3D exact solutions, considering multilayered piezoelectric composite plates of different aspect ratios, under an applied transverse load or surface potential. It is shown that both models are able to predict an accurate quasi-3D description of the static electromechanical analysis of multilayered plates for all aspect ratios.

Three-dimensional free vibration of multi-layered piezoelectric plates through approximate and exact analyses

This article presents two models that have the aim of analysing three-dimensional freely vibrating plates made of an arbitrary combination of structural and/or piezoelectric layers. The first model is derived from a displacement-based variational statement, and it investigates the possibility of approaching exact three-dimensional results at any degree of accuracy. This model has been developed as if the plates were virtually made of a single layer, and it is herein referred to as the approximate analysis model. The second model is based on solving the set of three-dimensional linear equations coupling the relevant mechanical and electric quantities, and it therefore provides exact results. The latter model is derived from the transfer-matrix technique which, having shown numerical instability in the multiphysics problem being dealt with, was then successfully modified to provide exact and reliable results. Excellent agreement has been obtained between the models, and this shows how the exact approach here designed is stably able to overcome ill conditioning problems, while the first model, having been validated by the exact results, could be applied to effectively investigate multiphysics problems for general boundary conditions, and for cross- and/or angle-ply laminates, at any level of required accuracy.

Free-ultrasonic waves in multilayered piezoelectric plates: An improvement of the Legendre polynomial approach for multilayered structures with very dissimilar materials

In 1999, Legendre polynomial approach has been proposed to solve wave propagation in the multilayered piezoelectric plate. However, it can deal with the multilayered plate only when the material properties of two adjacent layers do not change significantly. In this paper, an improvement of the Legendre polynomial approach is promoted to solve wave propagation in multilayered piezoelectric plates whatever the dissimilarities of the layer material properties. Detailed formulations are given to highlight the differences from the conventional Legendre polynomial approach. The validity of the proposed improved approach is illustrated through a numerical comparison between the improved method’s results and the exact solution obtained from the reverberation-ray matrix method. It is shown that the conventional orthogonal polynomial approach cannot conceptually calculate accurately, neither the discontinuous distribution of the normal electric field nor the continuous distributions of the normal stress and normal electric displacement, even in multilayered plates with similar layer material properties while the proposed improved polynomial approach overcomes these major drawbacks. Finally, the influences of the stacking sequences and volume fractions on dispersion curves are illustrated. It is also found that the stress and electric displacement of high frequency waves always distribute on the layer with lower wave speed.

Wave propagation in multilayered piezoelectric spherical plates

This paper proposes an improvement of the Legendre polynomial series method to solve the harmonic wave propagation in multilayered piezoelectric spherical plates, which are used in point-focusing transducers. The conventional Legendre polynomial method can deal with the multilayered structures only when the material properties of two adjacent layers do not change significantly and cannot obtain correctly normal stress and normal electric displacement shapes unlike the proposed improved orthogonal polynomial approach which overcomes these drawbacks. Detailed formulations are given to highlight its differences from the conventional Legendre polynomial approach. Through the comparisons of numerical results given by an exact solution (obtained from the reverberation-ray matrix method), and by the conventional polynomial approach and the improved polynomial approach, the validity of the proposed approach is illustrated. The influences of the radius-to-thickness ratio on dispersion curves, stress and electric displacement distributions are discussed. It is found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: radius-to-thickness ratio, wave speed, and position of the component material.

Wave propagation in the circumferential direction of general multilayered piezoelectric cylindrical plates

The Legendre polynomial approach has been proposed to solve wave propagation in multilayered flat plates and functionally graded structures for more than ten years, but it can deal with a multilayered plate only when the material properties of two adjacent layers do not change significantly. In this paper, an improvement of the Legendre polynomial approach is proposed to solve wave propagation in what, from now on, we will call general multilayered piezoelectric cylindrical plates, to mean indifferently with or without very dissimilar materials. Detailed formulations are given to highlight the differences from the conventional Legendre polynomial approach. Through numerical comparisons among the exact solution (from the reverberation-ray matrix), the conventional polynomial approach, and the improved polynomial approach, the validity of the proposed approach is illustrated. Then, the influences of the radius-to-thickness ratio on the dispersion curves, the stress, and electric displacement distributions are discussed. It is shown that the conventional orthogonal polynomial approach cannot obtain correct continuous normal stress and normal electric displacement shapes, unlike the improved orthogonal polynomial approach, which overcomes these drawbacks. It is also found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: the radius-to-thickness ratio, the wave speed of component material, and the position of the component material.

Coupled Thermo-Electro-Mechanical Analysis of Smart Plates Embedding Composite and Piezoelectric Layers

This article considers the static analysis of multilayered piezoelectric plates subjected to mechanical pressures and to field loads such as electric potential and temperature. The Principle of Virtual Displacements has been extended to thermo-electro-mechanical cases by including the virtual internal electric and thermal works and by writing the opportune constitutive relations. The governing equations are given in terms of three main primary variables (displacements, electric potential and temperature) and they are solved in a closed-form solution. The primary variables are expanded in the thickness direction in both Equivalent Single Layer (ESL) and Layer-Wise (LW) form with higher orders of expansion. The models proposed enable to obtain a quasi-3D description of the static thermo-electro-mechanical analysis of multilayered plates and to analyze the effects of each physical field involved. The limitations of classical theories for such analyses have also been discussed.

Assessment of a layerwise mixed least-squares model for analysis of multilayered piezoelectric composite plates

A layerwise mixed finite element model is developed based on the least-squares formulation for the coupled electromechanical static analysis of multilayered plates with piezoelectric and composite layers. The model assumes a layerwise variable description for displacements, transverse stresses and in-plane strains, along with the electrostatic potential, transverse electric displacement and in-plane electric field components, taken as independent variables. This original choice for the layerwise mixed formulation is intended to ensure the a priori and complete fulfilment of the interlaminar C0 continuity of both mechanical and electrical variables, which is due to compatibility and equilibrium conditions. Numerical applications are shown for assessment of the model predictive capabilities by comparison with available exact three-dimensional solutions, considering multilayered piezoelectric composite plates of various side-to-thickness ratios, under an applied load or surface potential.

Theoretical Analysis of Electromechanical Coupling Coefficient of Lamb Waves in ZnO/Si Multilayered Piezoelectric Plates

Lamb wave devices have been widely used in electro-acoustic and microfluidic devices. In order to improve their performances, the phase velocity dispersion and electromechanical coupling coefficient (ECC) of the Lamb wave should be calculated exactly during designing. Accordingly, this paper aims at analyzing exactly Lamb waves in multilayered piezoelectric plates with distinct electrode arrangements. First, the formulae of effective permittivity were derived based on the transfer matrix method and further was employed to calculate the phase velocity dispersion. The ZnO/Si multilayered plate was taken as the calculation example. The ECCs under distinct electrical boundary conditions were calculated by the Green's function method. Finally, the influences of the silicon thickness on the phase velocity dispersion and ECC are further discussed. Results show that the coupling coefficients deeply depends on the electrode arrangement, and the S0 mode with the electrode arrangements of type D is a better due to its larger velocity and higher coupling coefficient. Moreover, the ECC can be enlarged by reducing the nonpiezoelectric membrane thickness.

Elastic guided wave propagation in a periodic array of multi-layered piezoelectric plates with finite cross-sections

In this study, we present a model study of guided wave dispersion and resonance behavior of an array of piezoelectric plates with arbitrary cross-sections. The objective of this work is to analyze the influence of the geometry of an element of a 1D-array ultrasound transducer on generating multi-resonance frequency so as to increase the frequency bandwidth of the transducers. A semi-analytical finite-element (SAFE) method is used to model guided wave propagation in multi-layered 1D-array ultrasound transducers. Each element of the array is composed of LiNbO 3 piezoelectric material with rectangular or subdiced cross-section. Four-node bilinear finite-elements have been used to discretize the cross-section of the transducer. Dispersion curves showing the dependence of phase and group velocities on the frequency, and mode shapes of propagating modes were obtained for different geometry consurations. A parametric analysis was carried out to determine the effect of the aspect ratio, subdicing, inversion layer and matching layers on the vibrational behavior of 1D-array ultrasound transducers. It was found that the geometry with subdiced cross-section causes more vibration modes compared with the rectangular section. Modal analysis showed that the additional modes correspond to lateral modes of the piezoelectric subdiced section. In addition, some modes have strong normal displacements, which may influence the bandwidth and the pressure field in front of the transducer. In addition, the dispersion curves reveal strong coupling between waveguide modes due to the anisotropy of the piezoelectric crystal. The effect of the matching layers was to cluster extensional and flexural modes within a certain frequency range. Finally, inversion layer is found to have a minor effect on the dispersion curves. This analysis may provide a means to analyze and understand the dynamic response of 1D-array ultrasound transducers.

A differential reproducing kernel particle method for the analysis of multilayered elastic and piezoelectric plates

A differential reproducing kernel particle method for the analysis of multilayered elastic and piezoelectric plates

An exact analysis of forced thickness-twist vibrations of multi-layered piezoelectric plates

This paper deals with the thickness-twist vibration of a multi-layered rectangular piezoelectric plate of crystals of 6 mm symmetry or polarized ceramics. An exact solution is obtained from the three-dimensional equations of linear piezoelectricity. The solution is useful to the understanding and design of composite piezoelectric devices. A piezoelectric resonator, a piezoelectric transformer, and a piezoelectric generator are analyzed as examples.