Functionally Graded Piezoelectric Material Plate Research Articles

Electro-mechanical vibro-acoustic characteristics of submerged functionally graded piezoelectric plates with general boundary conditions

The present paper focuses on studying the underwater electro-mechanical vibro-acoustic characteristics of functionally graded piezoelectric material (FGPM) plates with general boundary conditions. For this purpose, the collocation points method is firstly developed to handle vibro-acoustic coupling, which can directly calculate the virtual work of secondary sound pressure via an easy double integral, without pre-simplifying the complex quadruple integral. The proposed FGPM plate model is established in the framework of first-order shear deformation theory (FSDT) and second type constitutive equations of piezoelectric materials, and the macroscopically inhomogeneous electro-mechanical properties are considered by means of Voigt model. The multi-physical fields including electric, mechanical and acoustic fields are uniformly extended in Chebyshev polynomials. Meanwhile, penalty function method is introduced to satisfy the arbitrary mechanical and electric constraints. All energy formulations are processed by Rayleigh-Ritz procedure and a standard governing equation is finally derived. Based on the proposed unified model, the convergence and accuracy for solving vibration and acoustic radiation of FGPM plates are validated firstly. Additionally, the influence of some key parameters on underwater vibro-acoustic characteristics of FGPM plates is fully examined. The results of this paper may provide a reference for future application of FGPM-based smart structures.

On impact process of FGPM plate with the damaged interlayer: piezoelectricity-based interaction effect analysis and stress field prediction using BPNN

Revealing the piezoelectricity-based interaction effect on the impact process of functionally graded piezoelectric material(FGPM) plates with the damaged interlayer promotes the intelligent development of engineering structures. The extended neural network algorithm is conducive to the integrated design of the intelligent structure. The impact process of FGPM plate with damaged interlayer is investigated. In the framework of Reddy's higher-order shear plate theory, the reduced constitutive equations considering the piezoelectric effect and damage effect are established for the FGPM layer and damaged interlayer respectively. The contact forces in the loading and unloading processes are modeled by the modified Hertz contact theory. The impact dynamic equations are derived through the Hamiltonian variational principle. The high-order truncated double trigonometric series solution in the impact process is extended. Further, the back propagation neural network(BPNN) technique is developed to predict the stress state during the impact process. Ultimately, from the perspective of energy conversion, the interaction of piezoelectricity and damage effect, structural parameters and loading state on the impact process are systematically revealed. The developed BPNN technique achieves accurate prediction of the flexural stress fields.

Nonlinear aeroelastic analysis and active flutter control of functionally graded piezoelectric material plate

The nonlinear aeroelastic characteristics and active flutter control of functionally graded piezoelectric material (FGPM) plate under thermo-electro-mechanical loads are studied. An analytical model is developed to explore the nonlinear aeroelastic behaviors of the supersonic FGPM plate, in which the nonlinear effects caused by geometry, aerodynamic force and piezoelectric effect are considered. Based on power law and sigmoid distribution, an improved sigmoid distribution is proposed for the aeroelastic design of the FGPM plate. Numerical examples are carried out to demonstrate that material volume fraction and gradient distribution can significantly affect flutter and thermal buckling of the FGPM plate. Additionally, the effects of thermal and electric loads on limit cycle oscillations and dynamic bifurcations of nonlinear FGPM plate are explored. Results indicate that the applied voltage can cause the static deflection of the plate and the thermal load can result in a more complex dynamic evolution process, and they reveal the competition mechanism of mechanical instabilities of the FGPM plate. Furthermore, the active flutter control of FGPM plate based on displacement feedback, actuated by FGPM plate itself, is carried for improving the aeroelastic stability of the system. The influence factors of active control are explored through parameter analysis, and results show that the volume fraction of piezoelectric material and thermal loads can significantly affect the active control effect. The present work demonstrates that the flutter stability of the FGPM plate can be improved by optimization of the material distribution and self-actuated active control.

Modeling of the Liouville–Green method to approximate the mechanical waves in functionally graded and piezo material with a comparative study

The present research article studies and compares the surface waves transmission through the functionally graded piezoelectric material (FGPM) club between the piezomagnetic (PM) layer -and half-space, and for a comparative study, lower half-space is assumed to be piezoelectric material. The transmission of mechanical waves in a smart structure is analyzed by following the elastic wave theory of magneto-electro-elasticity. The Liouville-Green (LG) approximation technique is used to solve the differential equation in the FGPM stratum, where exponential variation is assumed in material gradients. It is noticed that the material gradients depend considerably on the angular frequency, which should be a crucial factor in regulating the dispersion characteristics of functionally graded materials (FGM) waveguides. In closed determinant form, the dispersion relation has been obtained for FGPM plate for electrically open and short cases. The profound effect of parameters, such as material gradient, a width of the layer on phase velocity, coupled electromechanical factor, and angular velocity, is observed and delineated graphically. Different parametric plots are sub-plotted into a single figure to increase the readability of the graphs.

Investigation of the Nonstationary Stochastic Response of Functionally Graded Piezoelectric Material Plates with General Boundary Conditions

The functionally graded piezoelectric material (FGPM) has been widely used in various engineering cases due to its excellent mechanical properties. So far, the investigations on the stochastic dynamic behaviors of FGPM structures are few. This paper develops a unified solution for the nonstationary stochastic responses of the FGPM plate with general boundary conditions, in which the point, distributed and base acceleration random excitations can be considered. The first-order shear deformation theory (FSDT) is adopted to formulate the energy functional of the system and the governing equations are deduced by using the Hamilton's principle. The solutions of the system are obtained based on the modified Ritz method and pseudo excitation method (PEM). The accuracy of the proposed model is validated by comparing the obtained free vibration and random response results with those from the published literature. Finally, the effects of key parameters including boundary conditions, external electric voltage, material constituent and time modulated function on the stochastic response of the FGPM plate are investigated.

Nonlinear electro-thermo-mechanical dynamic behaviors of a supersonic functionally graded piezoelectric plate with general boundary conditions

A unified analytical model is developed to investigate the nonlinear aeroelastic behaviors of a supersonic functionally graded piezoelectric material (FGPM) plate with general boundary conditions under electro-thermo-mechanical loads. The formulation is derived by first-order shear deformation theory (FSDT) and supersonic piston theory, and the geometrical nonlinearity is considered based on von Karman large deformation theory. The motion equations of the supersonic FGPM plate are obtained through Hamilton principle and a modified Fourier series with auxiliary functions is employed to satisfy the possible mechanical and electric boundary conditions. Nonlinear aeroelastic responses of the FGPM plate are solved via Newmark integration technique combined with Newton-Raphson iterative scheme. Convergence and comparison studies show that the proposed model has sufficient accuracy in predicting aeroelastic stability and nonlinear dynamic responses as well as possess reliability in handling arbitrary boundary conditions. Numerical examples are carried out to demonstrate that several key parameters can significantly affect flutter and thermal buckling of the plate. Additionally, the effects of thermal and electric loads on limit cycle oscillation and dynamic bifurcation of nonlinear FGPM plate are examined. A higher temperature rise can result in the transition of the system from stable to chaos and more complicated evolution of dynamic motions.

Bending behaviors of the in-plane bidirectional functionally graded piezoelectric material plates

The transverse bending behaviors of in-plane bidirectional functionally graded piezoelectric material (FGPM) plates are semi-analytically investigated by the scaled boundary finite element method (SBFEM) in association with the precise integration method (PIM). The proposed scheme is able to explore the structural characteristics of FGPM plates with the material coefficients obeying arbitrary form of mathematical functions according to the in-plane coordinates. The present methodology selects only four quantities consisting of three translational displacement components and the electric potential as the fundamental unknowns. Additionally, variations of the four primary variables across the thickness direction are expressed as an analytical exponential matrix. In the developed approach, plates are regarded as a kind of three dimensional structure. But, an arbitrary in-plane surface discretized with two-dimensional spectral elements is set as the research domain. The practice is conducive to cut down the computational effort and increase the calculation efficiency. The SBFEM governing equations are formulated from the three-dimensional basic equations of piezoelectric materials without any assumptions on the plate kinematics and distributions of electromechanical components. By means of the scaled boundary coordinate system and the dual vector methodology, the key partial differential equations of piezoelectric materials are conveniently converted into the easily-solved first order ordinary differential SBFEM governing equation. The stiffness matrix is constructed from the analytical exponential matrix aided by the highly accurate PIM to predict the changing patterns of mechanical and electric quantities. To further improve the precision, the technology of dividing the plate into two parts with equal thickness is exploited. Finally, numerical exercises of square, rectangular and triangular piezoelectric plates are provided to validate the accuracy and fast convergence of the developed technique and reveal the effect of geometrical shapes, gradient functions, types of external loadings and thickness-to-span ratios on the static flexure of FGPM plates owning the in-plane bidirectional stiffness.

Flutter and Thermal Buckling Properties and Active Control of Functionally Graded Piezoelectric Material Plate in Supersonic Airflow

This paper is devoted to investigate the flutter and thermal buckling properties of the functionally graded piezoelectric material (FGPM) plate in supersonic airflow, and the active flutter control is carried out under different temperature fields. The piezoelectric material component of the FGPM plate has gradient changes along the thickness, such as piezoelectricity and dielectric coefficients. The supersonic piston theory is used to evaluate the aerodynamic pressure. Based on the first-order shear deformation theory and Hamilton’s principle with the assumed mode method, the equation of motion of the structural system is deduced. The effect of aerodynamic pressure on the frequency and damping ratio of the FGPM plate is analyzed. Moreover, the flutter and thermal buckling properties of the FGPM and pure metal plates are compared to show the superior aerothermoelastic properties of the FGPM plates. The influences of volume fraction exponent and temperature on the flutter and thermal buckling properties of the FGPM plate are also examined in detail. The LQR controller is adopted to achieve active flutter control. The simulation results show that the present control method can largely improve dynamic stability of the FGPM plate in supersonic airflow and high-temperature environment.

Modal analysis of functionally graded piezoelectric material plates

The goal of this research is to execute a modal analysis of the functionally graded piezoelectric material rectangular plate to find the natural frequency with different mode shapes by using the Finite Element (FE) Technique. The functionally graded piezoelectric material plate is considered to be graded along the thickness and follow the simplified power-law. FEM code has been developed in ANSYS 17.2 for modal analysis of the functionally graded piezoelectric material (FGPM) rectangular plate. The FGPM Rectangular plate mode shapes and natural frequencies are observed at varying boundary conditions. The effect of power-law exponent n variation on the natural frequency of the FGPM plate with different boundary conditions are considered, and results are presented.

Active Control of Plates Made of Functionally Graded Piezoelectric Material Subjected to Thermo-Electro-Mechanical Loads

The present paper concentrates on the active control of the static bending and dynamic response of a functionally graded piezoelectric material (FGPM) plate subjected to thermo-electro-mechanical loads. Using the first-order shear deformation theory and Hamilton’s principle, the equation of motion for the FGPM plate is deduced. Based on the smart properties of piezoelectric ceramic, the active control of the static bending and vibration of the FGPM plate is studied by the mechanical load feedback, velocity feedback and LQR control methods in thermal environment. The effects of load parameter, temperature, volume fraction index and feedback control gain on the static bending and dynamic response of the FGPM plate are examined in detail. The simulation results show that the present control method can largely improve both the static and dynamic stability of the FGPM plate.

Active vibration control of functionally graded piezoelectric material plate

The present paper is concerned with active vibration control of functionally graded piezoelectric material (FGPM) plate using the piezoelectric material component as actuator. Using the classical laminated plate theory, the equation of motion for the FGPM plate is deduced based on Hamilton’s principle and Rayleigh-Ritz method. A velocity feedback control method is used to obtain an effective active damping in the vibration control. The influences of distribution type, volume fraction index and total volume fraction of piezoelectric material on the vibration control of FGPM plate are investigated. The calculation results show that the vibration control result is strongly affected by the distribution of piezoelectric materials in FGPM plate. It is also found that the total volume fraction, especially the volume fraction index play an important role in the vibration control. Furthermore, under the non-uniform electric field the effect of external voltage position on active vibration control is studied. The results show that one can obtain an excellent control effect by optimizing the structure of FGPM plate and the position of external control voltage.

Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state

To provide reference for aerospace structural design, electro-mechanical vibrations of functionally graded piezoelectric material (FGPM) plates carrying porosities in the translation state are investigated. A modified power law formulation is employed to depict the material properties of the plates in the thickness direction. Three terms of inertial forces are taken into account due to the translation of plates. The geometrical nonlinearity is considered by adopting the von Kármán non-linear relations. Using the d’Alembert's principle, the nonlinear governing equation of the out-of-plane motion of the plates is derived. The equation is further discretized to a system of ordinary differential equations using the Galerkin method, which are subsequently solved via the harmonic balance method. Then, the approximate analytical results are validated by utilizing the adaptive step-size fourth-order Runge-Kutta technique. Additionally, the stability of the steady state responses is examined by means of the perturbation technique. Linear and nonlinear vibration analyses are both carried out and results display some interesting dynamic phenomenon for translational porous FGPM plates. Parametric study shows that the vibration characteristics of the present inhomogeneous structure depend on several key physical parameters.

Liouville-Green approximation: An analytical approach to study the elastic waves vibrations in composite structure of piezo material

This research article delves the study of surface waves in functionally graded piezoelectric material (FGPM) clubbed between two dissimilar piezomagnetic (PM) media. The transference of elastic waves in a composite structure is analyzed following the elastic wave theory of magneto-electro-elasticity. Liouville-Green’s (LG) approximation technique is used to solve the differential equation. The exponential variation is assumed in material gradients of FGPM stratum. It is noticed that the frequency of considered wave depends significantly on the material gradient coefficients, which may be a crucial factor to regulate the dispersion characteristics of functionally graded material (FGM) waveguides. Frequency equations have been obtained for electrically open and short cases in determinant form. The profound effect of parameters like material gradients (piezoelectric, dielectric and elastic) and width of the layers, on the phase velocity of Love type wave are presented graphically. Moreover, it is noticed that the material gradients also influences the electromechanical coupling factor. This influence has shown through the graph. Different parametric curves are merged into a single figure to increase the readability of the graphs. The magnetic potential function is derived analytically for all three gradient factors of FGPM plate. Obtained results are matched analytically and graphically with the established results.

Porosity-dependent nonlinear forced vibration analysis of functionally graded piezoelectric smart material plates

This work investigates the porosity-dependent nonlinear forced vibrations of functionally graded piezoelectric material (FGPM) plates by using both analytical and numerical methods. The FGPM plates contain porosities owing to the technical issues during the preparation of FGPMs. Two types of porosity distribution, namely, even and uneven distribution, are considered. A modified power law model is adopted to describe the material properties of the porous FGPM plates. Using D’Alembert’s principle, the out-of-plane equation of motion is derived by taking into account the Kármán nonlinear geometrical relations. After that, the Galerkin method is used to discretize the equation of motion, resulting in a set of ordinary differential equations with respect to time. These ordinary differential equations are solved analytically by employing the harmonic balance method. The approximate analytical results are verified by using the adaptive step-size fourth-order Runge–Kutta method. By means of the perturbation technique, the stability of approximate analytical solutions is examined. An interesting nonlinear broadband vibration phenomenon is detected in the FGPM plates with porosities. Nonlinear frequency-response characteristics of the present smart structures are investigated for various system parameters including the porosity type, the porosity volume fraction, the electric potential, the external excitation, the damping and the constituent volume fraction. It is found that these parameters have significant effects on the nonlinear vibration characteristics of porous FGPM plates.

Coupled electro-elastic analysis of functionally graded piezoelectric material plates

A unified formulation of finite layer methods (FLMs), based on the Reissner mixed variational theorem (RMVT), is developed for the three-dimensional (3D) coupled electro-elastic analysis of simply-supported, functionally graded piezoelectric material (FGPM) plates with open- and closed-circuit surface conditions and under electro-mechanical loads. In this formulation, the material properties of the plate are assumed to obey an exponent-law varying exponentially through the thickness coordinate, and the plate is divided into a number of finite rectangular layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the primary field variables of each individual layer, respectively, such as the elastic displacement, transverse shear and normal stress, electric potential, and normal electric displacement components. The relevant orders used for expanding these variables in the thickness coordinate can be freely chosen as the linear, quadratic and cubic orders. Four different mechanical/electrical loading conditions applied on the top and bottom surfaces of the plate are considered, and the corresponding coupled electro-elastic analysis of the loaded FGPM plates is undertaken. The accuracy and convergence rate of the RMVT-based FLMs are assessed by comparing their solutions with the exact 3D piezoelectricity ones available in the literature.

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.

Static and dynamic analysis of functionally graded piezoelectric plates under mechanical and electrical loading

Analysis of the static bending, free vibration and dynamic response of functionally graded piezoelectric plates has been carried out by the finite element method under loadings. The FGPM (Functionally Graded Piezoelectric Material) plate is assumed to be graded through the thickness and simple power law distribution, in terms of the volume fractions of the constituents used for formulation. The electric potential is assumed linear through the thickness of the plate. The governing equations are obtained, using potential energy and Hamilton’s principle, based on the First order Shear Deformation Theory (FSDT), which can include thermo-piezoelectric effects. The finite element model is derived based on the constitutive equation of the piezoelectric material, accounting for coupling between the elasticity and electric effect, by four node elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plates under different mechanical boundary conditions. Numerical results for a PZT-4/PZT-5H plate are given in both dimensionless tabular and graphical forms. Effects of material composition and boundary conditions on static bending, free vibration and dynamic response are also studied. The numerical results obtained by the present model are in good agreement with the solutions reported in the literature.

Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates

An inhomogeneous layer element method is presented to analyze the dispersion of waves and characteristic wave surfaces in plates of functionally graded piezoelectric material (FGPM). In this method, the FGPM plate is divided into a number of layered elements. The elemental elastic and electric properties are assumed as linear functions of the thickness to adopt the variety of the material property of FGPM. The Hamilton principle is applied to determine the governing equations. The phase velocity surface, phase slowness surface, phase wave surface, group velocity surface, group slowness surface, and group wave surface for FGPM plate are formulated using Rayleigh quotient and the orthogonality condition of the eigenvectors. These six surfaces are then used to illustrate the characteristics of waves in FGPM plates. Numerical examples are presented using the present formulations to analyze dispersions and characteristics of waves in FGPM plates.