Functionally Graded Piezoelectric Research Articles

Nonlocal strain gradient-based dynamic stability of functionally graded piezoelectric nanoshells considering thermo-electro-mechanical coupling effects

In this paper, a nonclassical shell model is utilized to investigate the dynamic stability of functionally graded piezoelectric (FGP) nanoshells considering the thermo-electro-mechanical coupling effects. Also, the FGP nanoshells on elastic foundations are subjected to periodic axial loading. The theoretical formulations are derived by implementing the first-order shear deformation (FSD) shell theory into the nonlocal strain gradient theory (NSGT). Hamilton’s principle is utilized to obtain the equations of motion and related boundary conditions of the present nanoshell model. Moreover, the Mathieu-Hill equations determining the instability regions are presented and analyzed based on Bolotin’s method. Results demonstrate that the external electric voltage, the elastic foundations, the temperature rise, the static load factor, the power-law index and the small-scale parameters have significant influences on the dynamic stability responses of FGP nanoshells.

The theory of thermoelasticity with a memory-dependent dynamic response for a thermo-piezoelectric functionally graded rotating rod

By laminating piezoelectric and flexible materials during the manufacturing process, we can improve the performance of electronic devices. In smart structure design, it is also important to understand how the functionally graded piezoelectric (FGP) structure changes over time when thermoelasticity is assumed. This is because these structures are often exposed to both moving and still heat sources during many manufacturing processes. Therefore, it is necessary to conduct theoretical and experimental studies of the electrical and mechanical characteristics of multilayer piezoelectric materials when they are subjected to electromechanical loads and heat sources. Since the infinite speed of heat wave propagation is a challenge that classical thermoelasticity cannot address, other models based on extended thermoelasticity have been introduced. For this reason, the effects of an axial heat supply on the thermomechanical behavior of an FGP rod using a modified Lord-Shulman model with the concept of a memory-dependent derivative (MDD) will be explored in this study. The exponential change of physical properties in the direction of the axis of the flexible rod will be taken into account. It was also assumed that there is no electric potential between the two ends of the rod while it is fixed at both ends and thermally isolated. Applying the Laplace transform method, the distributions of the physical fields under investigation were calculated. The obtained results were compared to those in the corresponding literature with varying heterogeneity values, kernel functions, delay times, and heat supply speeds. It was discovered that the studied physical fields and the dynamic behavior of the electric potential are weakened by increasing the inhomogeneity index.

Multilevel Fully Integrated Electromechanical Property Modulation of Functionally Graded Graphene‐Reinforced Piezoelectric Actuators: Coupled Effect of Poling Orientation

AbstractThis article explores the coupled static and dynamic electromechanical responses of single and multilayered functionally graded (FG) graphene platelet (GPL)‐reinforced piezoelectric composite (GRPC) plates by developing a 3D finite‐element model. The bending and eigenfrequency of piezoelectric FG composite plates are investigated, wherein an active behavior is proposed to be exploited in terms of the functional design of poling angle for a more elementary level property modulation. The numerical results reveal that the mechanical behavior concerning deflection and resonance frequency of FG‐GRPC plates can be significantly enhanced and modulated due to the influence of piezoelectricity and a small fraction of GPLs along with the consideration of poling angle in a multiscale fully integrated computational framework. The notions of on‐demand property modulation, actuation, and active control are established here by undertaking a comprehensive numerical analysis considering the coupled influences of poling orientations, different distributions, patterns, and weight fractions of GPLs along with different electromechanical loadings. Against the backdrop of the recent advances in microscale manufacturing, the current computational work will provide necessary physical insights in modeling piezoelectric multifunctional FG composites for active control of mechanical properties and harvesting electromechanical energy in a range of devices and systems.

Wave propagation analysis of porous functionally graded piezoelectric nanoplates with a visco-Pasternak foundation

This study investigates the size-dependent wave propagation behaviors under the thermoelectric loads of porous functionally graded piezoelectric (FGP) nanoplates deposited in a viscoelastic foundation. It is assumed that (i) the material parameters of the nanoplates obey a power-law variation in thickness and (ii) the uniform porosity exists in the nanoplates. The combined effects of viscoelasticity and shear deformation are considered by using the Kelvin-Voigt viscoelastic model and the refined higher-order shear deformation theory. The scale effects of the nanoplates are captured by employing nonlocal strain gradient theory (NSGT). The motion equations are calculated in accordance with Hamilton’s principle. Finally, the dispersion characteristics of the nanoplates are numerically determined by using a harmonic solution. The results indicate that the nonlocal parameters (NLPs) and length scale parameters (LSPs) have exactly the opposite effects on the wave frequency. In addition, it is found that the effect of porosity volume fractions (PVFs) on the wave frequency depends on the gradient indices and damping coefficients. When these two values are small, the wave frequency increases with the volume fraction. By contrast, at larger gradient index and damping coefficient values, the wave frequency decreases as the volume fraction increases.

Bending and buckling analysis of functionally graded Timoshenko nanobeam using Two-Phase Local/Nonlocal piezoelectric integral model

Previous studies indicate that nonlocal piezoelectric differential model would lead to an inconsistent bending response of Euler-Bernoulli nanobeam. In this paper, static bending and elastic buckling of functionally graded piezoelectric (FGP) Timoshenko nanobeam are studied through two-phase local/nonlocal piezoelectric integral model. The differential governing equations and standard boundary conditions are derived on the basis of the principle of minimum potential energy. The relations between general strain and nonlocal stress components are expressed as integral forms and then transformed equivalently into differential forms with constitutive boundary conditions. Several nominal variables are introduced to simplify the mathematical formulation. It shows that Timoshenko nanobeam based on purely nonlocal piezoelectric integral model would lead to ill-posed mathematical model and inconsistent size-dependent response with and without considering constitutive boundary conditions, respectively. The general differential quadrature method is applied to obtain the numerical results for bending deflections and buckling loads. The influence of nonlocal parameters, boundary conditions, functionally graded index and buckling order on the bending deflections and buckling loads is investigated numerically. A consistent softening response is obtained.

New finite modelling of the nonlinear static bending analysis of piezoelectric FG sandwich plates resting on nonlinear elastic foundations

This paper presents a new finite modelling of the nonlinear static bending analysis of piezoelectric functionally graded (FG) sandwich plates resting on nonlinear elastic foundations. Finite element formulations are derived by using the first-order shear deformation theory (FSDT) of Mindlin and the finite element method. The proposed theory and mathematical model of this work are verified by comparing the results with those of other methods, and they are in good agreement. A parameter study is conducted to investigate the effects of geometrical and physical properties such as nonlinear factors, volume fraction index, nonlinear foundation parameters, applied voltages, boundary conditions, etc., on the nonlinear mechanical behaviors of piezoelectric FG sandwich plates. The novel numerical results of this work are very important, and can be used as a good reference to examine related structures as well as of use in engineering practice.

Elastic Buckling and Free Vibration of Functionally Graded Piezoelectric Nanobeams Using Nonlocal Integral Models

Previously it was shown that nonlocal piezoelectric differential models will lead to inconsistent bending response for Euler–Bernoulli beams under different loadings and boundary conditions. In this paper, the general strain- and stress-driven two-phase local/nonlocal integral piezoelectric models are applied to analyze the elastic buckling and free vibration of functionally graded piezoelectric (FGP) Euler–Bernoulli beams under different boundary conditions. The differential governing equations and standard boundary conditions are derived by the Hamilton’s principle. The relations between the general strain and general nonlocal stresses are expressed as integral equations, which are further transformed equivalently to differential forms with constitutive boundary conditions. Several nominal variables are introduced to simplify the differential governing and constitutive equations, as well as the standard and constitutive boundary conditions. The general differential quadrature method is applied to obtain the numerical results of buckling loads and vibration frequencies of the FGP Euler–Bernoulli beam. Numerical results show that general strain- and stress-driven two-phase local/nonlocal integral piezoelectric models will lead to consistently softening and toughening size-dependent responses, respectively.

Stability Analysis of Smart FG Sandwich Plates with Auxetic Core

As a new model, functionally graded piezoelectric (FGP) sandwich plate with negative Poisson’s ratio honeycomb core (auxetic core) is considered in this paper. Buckling analysis of the FGP sandwich plate is investigated based on a novel four-unknown shear deformation plate theory. The electrical and mechanical properties of the face layers are continuously varied through the thickness of the layers. This variation is achieved using a power law distribution in terms of the constituents volume fraction. The core layer composed of hexagonal honeycomb cells with negative Poisson’s ratio was made of a metallic material. The sandwich plate is exposed to uniaxial or biaxial compressive loads as well as electric voltage. Five stability differential equations are established based on the principle of virtual work including mechanical and electric loads. The obtained buckling load is compared with that available in the literature. Impacts of various parameters like the power law index, load parameter, external applied voltage, core thickness, boundary conditions and plate geometry on the buckling load of the smart composite plates with auxetic core are investigated. From the numerical results, one can find that the increase of electric voltage and core thickness decreases the buckling load.

Analytical buckling temperature prediction of FG piezoelectric sandwich plates with lightweight core

Buckling temperature prediction of a functionally graded piezoelectric (FGP) sandwich plate with lightweight core is studied in the present work. The composite plate is made of three layers. The upper and lower layers are made of two different piezoelectric materials so that the electrical and mechanical properties are smoothly varied through the thickness based on a power law distribution. Whereas, the lightweight core is considered as hexagonal honeycomb structure or functionally graded porous structure. The porous core contains internal pores with different porosity distributions. According to a modified four-unknown shear deformation plate theory, the displacement field is described. The smart advanced plate is subjected to thermal and humid loadings as well as external electric voltage. The thermal and humid loadings may be uniform, linear or non-linear through the thickness. The stability equations are constructed from the principle of virtual work based on the proposed shear deformation plate theory. The obtained results will be validated by introducing some comparison examples. Finally, the influences of different parameters like the side-to-thickness ratio, aspect ratio, core type, core thickness, power law index, moisture concentration, external applied voltage and boundary conditions on the buckling temperature change of the sandwich plates with honeycomb core or with porous core are discussed. It is found that the buckling temperature of the sandwich plate with honeycomb core is greater than that of the sandwich plate with porous core. Moreover, with increasing the lightweight core thickness, the sandwich plate stiffness decreases leading to a reduction of the buckling temperature.

Vibration response analysis of sigmoidal functionally graded piezoelectric (FGP) porous plate under thermo-electric environment

In the present work, the vibration response of the porous functionally graded piezoelectric plates with electro-thermal loading using finite element formulations has been studied. A sigmoid law is used for the material property deviation along the thickness direction. The first-order shear deformation theory (FSDT) with von Karman nonlinear strains and Hamilton's principle is used to obtain the governing equations. The governing equations are solved by finite element methods with 9-noded iso-parametric rectangular elements with 7 degrees of freedom (DOFs) per node. The accuracy of the results is evaluated by comparing them with the results available in the literature. The analysis shows that the uneven type porosity distribution gives a larger nondimensional frequency than the even type porosity distribution FGP plate. The non-dimensional frequency decreases with increases in the ratio. Among all the conditions, CCCC boundary conditions have a larger frequency than SSSS, SFSF, SFCF, and CFFF. The obtained frequency under applied electric loading depends on the magnitude and the applied voltage sign. As the temperature change increases with the porosity exponent of the FGP plate, the nondimensional frequency is also increasing. The obtained results can design a porous FGP material-based smart composite structure under a thermo-electric environment.

Vibration response analysis of PZT-4/PZT-5H based functionally graded tapered plate subjected to electro-mechanical loading

In this paper, the vibration responses of the tapered functionally graded piezoelectric (FGP) square plate are analyzed. The governing equations are obtained using the first-order shear deformation theory (FSDT) with the variational (Hamilton's) principle, and the solution is obtained using nine-noded isoperimetric elements based finite element method. The effective electrical and mechanical properties are varying power-law distribution along the thickness direction. The tapered FGP plate is composed of PZT-4 (Top-side) and PZT-5H (bottom-side). Convergence and comparison tests have been carried out to validate the exactness of the current method. The numerical results are presented to analyze the effect of side to thickness(a/h)ratio, tapered ratio(ℏ), power law exponent(N), applied electric voltage(V)and the different boundary conditions for the tapered FGP square plate. The results show that the FGP tapered plate frequencies are influenced by the tapered ratio, volume fraction exponent, side to thickness ratio, and the applied electric voltage. These findings can be used to design smart structures based on the functionally graded piezoelectric material.

An efficient scaled boundary finite element approach in bending and bucking analysis of functionally graded piezoelectric plates

Based on the three-dimensional (3D) linear elastic theory without enforcing any plate assumptions, the scaled boundary finite element method (SBFEM) is proposed to solve the solution of the displacements, electrical potentials, critical buckling loads and buckling modes in functionally graded piezoelectric (FGP) plate. The electro-elastic properties are realized by transitioning material phase in the form of exponential function across the thickness direction. According to the constitutive equation of FGP materials and considering the coupling of elastic and electrical effects, the SBFEM model is established. The principle of virtual work in conjunction with the Green's theorem is applied to obtain the SBFEM governing equations with respect to the displacement domains and piezoelectric effects. By placing the scaling center of the SBFEM at infinity, the two-dimensional (2D) mesh can be translated along the thickness to obtain the 3D plate geometry, which can reduce the dimension of the model as well as calculation cost and greatly improve the computational efficiency. The bottom surface of the FGP plate is discretized by 2D higher-order spectral elements because of their high accuracy and fast convergence speed. The radial solution can be expressed analytically as a matrix exponential function, which can be solved accurately by the Padé expansion, and a simple and effective stiffness matrix can be obtained. By comparing several numerical examples with the reference solutions, the results indicate that the present analysis has good accuracy and rapid convergence. It is worth mentioning that this method is free of shear-locking and is applicable in the case of deformed grids. The influences of geometric variables, material composition, boundary conditions as well as load forms on the bending and bucking response are investigated.

Characteristic orthogonal polynomials-Ritz method for vibration behavior of functionally graded piezoelectric plates using FSDT

This work focuses on the free vibration characteristics of the functionally graded piezoelectric (FGPE) plate with classical and elastic constraints. First, the analytical model is established for the FGPE plate on the basis of the first-order shear deformation theory (FSDT). Besides, different distribution profiles of the material constituent are considered for the FGPE plate model, and various external initial voltages are also considered for modeling the piezoelectric effect. The displacement functions for the FGPE plate are then given in the form of third-kind orthogonal polynomials. Ultimately, the solution of the FGPE plate model is resolved by means of the Ritz method. Convergence and accuracy of the presented model are verified, and a series of parametric investigations are given.

Nonlinear vibration behavior of doubly-curved functionally graded piezoelectric microshells in thermal environments

In this paper, nonlinear vibrational behaviors of doubly-curved functionally graded piezoelectric (FGP) microshells in thermal environments have been studied. The doubly-curved microshell has been modeled by using modified couple stress and thin shell theories. The microshell has two material constituents which are piezoelectric materials and their gradation is described using power-law functions. Temperature fields are considered to be uniform, linear, and nonlinear. A closed-form of nonlinear vibration frequency has been obtained by analytically solving the governing equations using the Harmonic balance method. It will be demonstrated that nonlinear vibrational frequency of doubly-curved microshell relies on couple stress influence, material gradation, curvature radius, center deflection, and electrical field.

The hygro-thermo-electro-mechanical coupling edge-based smoothed point interpolation method for the response of functionally graded piezoelectric structure under hygrothermal environment

In this paper, we analyze the hygro-thermo-electro-mechanical (HTEM) coupling problem of functionally graded piezoelectric (FGPE) structure based on edge-based smoothed point interpolation method (ES-PIM). The basic equations for FGPE structure are derived in the multi-physical field. Triangular elements adopted in ES-PIM can be generated automatically and suitable for complicated structures. The responses for the HTEM coupling problem are calculated by ES-PIM and finite element method (FEM) under different hygrothermal conditions. Through the numerical examples, the factors related to the temperature and moisture fields are explored. The exponential factor and the hygrothermal effect on the responses of the FGPE structure are also studied. Besides, the advantages of proposed method for HTEM coupling problems are also validated by several numerical examples. These solutions also indicate that the increase of temperature and moisture concentration decreases the structural stiffness when empirical constants are set to negative numbers, leading to the decrease of frequency and variation of displacement. This article proposes an excellent numerical method and provides a reference for the design and application of piezoelectric structures.

Analysis of Lamb wave propagation in a functionally graded piezoelectric small-scale plate based on the modified couple stress theory

This work focuses on the propagation characteristics of Lamb-type waves in a functionally graded piezoelectric (FGP) small-scale plate. Based on the modified couple stress theory, Legendre orthogonal polynomial method (LOPM) is extended to perform the solving process. By introducing the rectangular window function and expanding the electric potential and displacement vectors to Legendre orthogonal polynomial series, the governing equations satisfying the electric-open and electric-short circuit boundary conditions are presented respectively. The solutions based on the global matrix method (GMM) are deduced to verify the correctness of this extended LOPM. Since there is no layered processing, the extended LOPM provides a more realistic analysis model for the FGP small-scale plate and has a high computational efficiency. Furthermore, the dispersion characteristics, stress and electromechanical coupling factor are investigated in details. Results show that the size and piezoelectric effect increase the phase velocity, and the piezoelectricity suppresses the influence of couple stress on the dispersion characteristics. The Lamb wave phase velocities of the first and second modes with different Ly always converge to two fixed values at different convergence speeds. The conclusions obtained can be applied to the design and dynamic response evaluation of composite small-scale structures.

Hybrid integral transform and p-version FEM for thermo-mechanical analysis of a functionally graded piezoelectric hollow cylinder subjected to asymmetric loads

As the first endeavor, a combination of fast Fourier transform and p-version of finite element method is proposed for electro-thermo-elastic analysis of a thick hollow cylinder under asymmetric thermal loadings. Especially in shells of revolution, the proposed FFT-pFE method is accompanied by a significant decrease in the computational costs. Due to the problem periodicity in such structures, the fast Fourier transform technique is used to discretize the governing equations into a set of harmonics in circumferential direction. Each harmonic is then partitioned using higher order finite elements. Hierarchical finite elements based on Legendre polynomial interpolation functions are utilized to discretize 2D governing equations of a functionally graded piezoelectric (FGP) cylinder. 3D governing equations of a FGP hollow cylinder are then discretized by using the higher-order Lagrangian finite elements. The effects of FFT grid-size as well as the order of the interpolation functions are investigated on convergence behavior of the proposed mixed FFT-pFE method. The piezoelectric material properties, with the exception of the Poisson’s ratio, are considered to vary along the radius of the cylinder and pursue the power function. The governing equations are derived using the principle of virtual displacements. For a 3D FGP hollow cylinder, the influence of ...

Application of nonlocal modified couple stress to study of functionally graded piezoelectric plates

In this paper, we study a size-dependent functionally graded piezoelectric (FGP) plate model based on the combination of the modified couple stress and nonlocal elasticity theories, named nonlocal modified couple stress theory (NMCS). Continues variation of the material properties (chosen from PZT family) through the thickness of the plate is considered according to the power law distribution. Employing Hamilton's principle via thin plate theory, we obtain governing equations of motion for two cases: 1) using only the nonlocal parameter for stress tensor, 2) using the nonlocal parameter for both stress and couple stress tensors. Navier's approach has been used to investigate free vibration of FGP under simply supported edge conditions, analytically. In the first step, we compare our results with the obtained results for the vibration of FGP in the previous studies using the modified couple stress theory and nonlocal elasticity theory, separately. Then, the effects of the aspect ratio, the length of the plate, the side to thickness ratio, FG power index, mode numbers, and the nonlocal and length scale parameters on normalized natural frequencies are investigated.

Size-dependent analysis of a functionally graded piezoelectric micro-cylinder based on the strain gradient theory with the consideration of flexoelectric effect: plane strain problem

In this study, a size-dependent analysis of functionally graded piezoelectric (FGP) micro-rotating cylinder is presented based on the plane strain condition and strain gradient theory, which is a non-classical theory capable of capturing the size effect in microscaled structures. The present model is used to analyze the FGP micro-rotating cylinder with the consideration of flexoelectric effects exposed to a symmetric magneto-electro-mechanical loading. All mechanical and electrical properties are assumed to be graded in the thickness direction according to a power-law distribution. With respect to the fifth-order strain gradient coefficient and electromechanical coupling, the constitutive equations are obtained from electric Gibbs free energy density, which is a function of strain, second-order deformation gradient and electric field. By substituting the constitutive equations in electric and mechanical equilibrium equations, two coupled electromechanical governing differential equations in terms of radial displacement and electric potential are derived considering centrifugal force and Lorentz magnetic force obtained from Maxwell’s relations. The generalized differential quadrature method is proposed to solve the coupled governing differential equations. Numerical results attained from the strain gradient elasticity reveal the effects of flexoelectric, microstructural length scale, non-homogeneity constant, rotation and magnetic field on the response of the FGP micro-rotating cylinder.

In-Plane Analysis of an FGP Plane Weakened by Multiple Moving Cracks

In this paper, the analytical solution of an electric and Volterra edge dislocation in a functionally graded piezoelectric (FGP) medium is obtained by means of complex Fourier transform. The system is subjected to in-plane mechanical and electrical loading. The material properties of the medium vary exponentially with coordinating parallel to the crack. In this study, the rate of the gradual change of the shear moduli and mass density is assumed to be same. At first, the Volterra edge dislocation solutions are employed to derive singular integral equations in the form of Cauchy singularity for an FGP plane containing multiple horizontal moving cracks. Then, these equations are solved numerically to obtain dislocation density functions on moving crack surfaces. Finally, the effects of the crack moving velocity, material properties, electromechanical coupling factor and cracks arrangement on the normalized mode I and mode II stress intensity factors and electric displacement intensity factor are studied.