Functionally Graded Piezoelectric Material Research Articles

Analytical approach on stress analysis in a functionally graded piezoelectric annular disk with varying compressibility and varying density under internal pressure

When a crystal is subjected to mechanical stress, it exhibits piezoelectricity, also known as the piezoelectric effect, which is the emergence of an electrical potential across the crystal's sides. A few examples of piezoelectric materials include quartz, Rochelle salt, lead zirconate titanate (PZT4) and zinc oxide (ZnO). Piezoelectricity has many applications in regards to electrical transducers and signal devices. In this paper, the analytical solution is obtained for elastic-plastic stresses in a thin rotating disc formed of functionally graded piezoelectric material with varying compressibility and varying density for distinct angular velocity with internal pressure using Seth's transition theory. The distribution of the mechanical and electrical components can be precisely controlled by applying functionally graded piezoelectric materials (FGPMs). Transition theory assumes that a mid-zone (transition state) exists between elastic state and plastic state. A nonlinear Governing equation of the problem is formulated using equilibrium equation by substituting the values of stresses. For evaluation of transitional stresses, transition function R is assumed in terms of radial stresses and equations are solved analytically by considering the transition (critical points) of the differential equation. The investigation focuses on how the rotational velocity and internal pressure affects the radial and circumferential distribution of stresses. With the help of graphs and mathematical calculations, it is observed that both variable density and variable compressibility have significant effect on the performance of a functionally graded piezoelectric materials. Therefore, it can be concluded that an annular disc made up of functionally graded piezoelectric materials is safer than that of homogeneous piezoelectric materials for engineering design purposes.

Thermo-electro-elastic coupled modeling and simulation of functionally graded piezoelectric structures

Compared with the composite materials, the functionally graded materials can not only provide better specific stiffness but also reduce the risk of layered damage. Therefore, the graded distribution forms have become the development trend of advanced structures. An efficient finite element (FE) modeling method and numerical simulation of plate and shell with nonlinear temperature gradient coupling are developed based on the FOSD hypothesis for thermo-electro-elastic coupled functionally graded piezoelectric material (FGPM) integrated with smart structures in this paper. The constitutive model for thermo-electro-elastic coupled FGPM is derived. Assuming that the temperature gradient distribution changes secondarily through the thickness, two nonlinear physical models for temperature change are constructed. Based on the FOSD hypothesis, the multi-field coupled FE equation of plate and shell is derived. The accuracy of the present FE model is verified by the commercial calculation structures. The parametric simulation of FGPM shells is carried out based on the FE model.

Static and natural frequency investigation of FGP beams considering thermal effects and design parameters

This study presents a static analysis and natural frequency analysis of functionally graded laminated piezoelectric beams based on the Euler–Bernoulli theory using the finite element method. A simple power law is used to vary all material properties across the thickness, except for Poisson’s ratio. The effect of laminate configuration and volume fraction index on the deflection and natural frequency of beams made of functionally graded piezoelectric materials (FGPM) is investigated, and the relationship between deflection and different volume fraction indices under thermal, electrical, and mechanical loads is explored. The study shows that there is a certain volume fraction index that maximizes or minimizes deflection. Additionally, the variation of natural frequency in relation to the power law index is examined. The findings of this research are useful for the development of sensors and actuators in different environments, and the appropriate operation point of the structure can be selected based on the behavior of the sensor or actuator of the beam.

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.

Scattering of Love-type wave in piezoelectric–piezomagnetic layered structure due to abrupt thickening

The present paper discusses the scattering phenomenon of Love-type wave on the functionally graded piezoelectric material (FGPEM) stratum overlying a functionally graded piezomagnetic material (FGPMM) substrate containing abrupt thickening at the uppermost free surface. The expressions for dispersion relation and group velocity (GV) are obtained for four kinds of electric-magnetic boundary conditions. The expressions for mechanical displacement (MD) component, electric potential (EP), and magnetic potential (MP) functions for stratum and substrate are deduced by using the perturbation method and residue theorem. The effect of piezoelectric and piezomagnetic coupling parameters and abrupt thickening parameters on phase velocity (PV), GV, MD component, and EP & MP functions associated with the stratum are graphically represented. A comparative study is carried out to delineate the concealed characteristics of wave on the considered model due to three different shaped abrupt thickening for four kinds of electric-magnetic boundary conditions (i.e. electrically short and magnetically short (ESMS), electrically open and magnetically short (EOMS), electrically open and magnetically open (EOMO), and electrically short and magnetically open (ESMO)).

Study of the dynamic electro-mechanical and microstructural behaviour on the elastic wave in FGM/FGPM composite structure: WKB asymptotic approach

This paper presents a theoretical investigation of the horizontally polarized shear wave in a composite structure. The assumed composite is composed of Functionally graded piezoelectric material (FGPM) layer constrained between an FGM (Functionally graded material) layer and a couple stress half-space with size-dependent microstructural effect. The variation in rigidities and densities are considered parabolic and linear in the upper and constrained layers, respectively. An analytical WKB (Wentzel–Kramers–Brillouin) asymptotic approach has been adopted for obtaining the dispersion relation. Also, a finite difference scheme (FDS) has been introduced to calculate the phase velocity and group velocity of the SH-wave. If waves travel in the time domain, it is necessary to check the accuracy and stability requirements. So that FDS has been adopted because of its accuracy, reliability, and flexibility. For the purpose of numerical calculation, three sets of FGPMs are considered namely, PZT−4, PZT−5H ceramic, and ceramic. Graphs have been plotted by means of dimensionless phase velocity against dimensionless wave number to show the effect of various parameters on the propagation of SH-wave. Numerical results demonstrated the accuracy and versatility of the phase and group velocity depending on the stability ratio of the FDS.

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.

Thermo-electro-mechanical dynamic and free vibration analysis of FGPM nanobeam with initial stress

Currently, nanostructures such as nanobeams, nanomembranes and nanoplates have been a hot area of research triggering. The contribution of this research article deals with both the effect of the free and dynamic vibration of a nanobeam as well as the use of the FGM material by integrating the piezoelectric effect. In fact, incorporating the initial stress for a nanoscale beam has not been considered yet. This paper displays an analysis of dynamic and free vibration of power-law functionally graded piezoelectric materials (FGPM) subjected to thermal-electro-mechanical loads with initial stress. The model was resolved based on nonlocal Timoshenko beams theory. The dimensionless fundamental frequency gradually improved with the new configuration of the FGPM nanobeam by 5% compared to the FGM nanobeam. The accuracy of this model is highlighted via numerical examples for different boundary conditions. To solve the governing equations, the polynomial differential quadrature method (PDQM) was elaborated and resolved numerically using MATLAB. For various configurations of the FGPM nanobeams, the effect of scaling parameters, volume fraction and slenderness ratios were discussed. In addition, for different edge conditions, variations of the first four frequencies with volume fraction indices and for different scaling parameters were set forward. This study corroborated that the power exponent, nonlocal parameter, slenderness ratio and thermo-electro-mechanical loads have significant effects on the dimensionless frequencies of various FGPM nanobeams. The present work provides an enlightening as guideline for the draft of a nanoscale beams based thermo-electro-mechanical actuators.

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.

SH wave vibration in functionally graded porous piezoelectric composite loaded with Newtonian conductive/non-conductive liquid

This study concentrates on the propagation pattern of shear horizontal (SH) waves on the conductive liquid-loaded porous piezoelectric media. Porous piezoelectric media is composed of fluid saturated functionally graded porous piezoelectric material (FGPPM) plate and functionally graded piezoelectric material (FGPM) half-space. The functional grading is assumed to be exponential for the FGPPM layer and quadratic for the FGPM layer respectively, along the direction of thickness. The mechanical imperfection is considered at the interface of the two media. The dispersion and the attenuation curve have been plotted to describe the impact of various parameters, namely gradient parameter of both the piezoelectric material, thickness of FGPPM plate, imperfect interface parameter, relative permittivity and conductivity of the liquid medium. Also, different modes for both the dispersion and attenuation curves have been delineated to get a better insight. Dispersion equations for non-conductive liquid loaded piezo-composite have also been obtained for both electrically open and short circuit conditions. To validate the present outcome, the geometry is reduced to special cases and the outcome is matched with pre-existing results. This article may find its application in better understanding of SAW devices incorporated with liquid layer.

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.

Static response of small scale functionally graded piezoelectric nanobeams

ABSTRACT In this paper, we tackle the static response of power-law functionally graded piezoelectric materials (FGPM) subjected to electro-mechanical stimulation. Effects of material composition, nonlocal parameter, slenderness ratio and electrical voltage on different characteristics of FGPM nanobeams are addressed. This study corroborates that the power exponent, nonlocal parameter and slenderness ratio significant impacts on reaction force of various configurations of FGPM nanobeams.

Elasticity Solutions for Cylindrical Bending of Functionally Graded Piezoelectric Plates

Functionally graded piezoelectric materials (FGPMs), combining the properties of functionally graded materials and piezoelectric materials, provides a new idea for multi-functional and intelligent lightweight components, and it has broad application prospects in electronic devices. Based on the elasticity and electric equilibrium equations, the Mian and Spencer functionally graded plate theory is extended from elastic materials to piezoelectric materials to study the cylindrical bending of FGPM plates, where the material elastic constants, piezoelectric and dielectric parameters are assumed to vary continuously and arbitrarily along thickness direction. Accordingly, the elasticity solutions for cylindrical bending of FGPMs plates under the uniform transverse loading are obtained. Numerical examples are given to demonstrate the influence of piezoelectric effects on the static responses of the presented FGPMs plates.

Transient Response of a Hollow Functionally Graded Piezoelectric Cylinder Subjected to Coupled Hygrothermal Loading

In this paper, transient response of a simply supported finite length hollow cylinder made of functionally graded piezoelectric material (FGPM) subjected to coupled hygrothermal loading was investigated. The coupled equations of heat conduction and moisture diffusion as well as motion equations and the electrostatic equation of FGPM were solved employing the Fourier series expansion method through the longitudinal direction, the differential quadrature method (DQM) along the radius and Newmark method for time domain. Finally, the distribution of temperature, humidity, electric potential, stresses and displacements was achieved. The effect of coupled and uncoupled hygrothermal loading, grading index and hygrothermal loading was illustrated in the numerical examples. The results show that using the coupled model is vital for analysis of transient response of the cylinder subjected to hygrothermal loading.

Thermoelastic contact problem of a rigid punch sliding on a functionally graded piezoelectric layered half plane with heat generation

In this study, the plane steady state thermoelastic frictional contact problem between a rigid cylindrical punch and a functionally graded piezoelectric material (FGPM) layer bonded to the homogeneous half plane is considered. The rigid punch is assumed to be a perfect electric conductor and thermal insulator and slides over the FGPM layer with a small constant speed and heat flux is generated due to the friction. Applying the Fourier integral transform technique and thermoelasticity, general stress, displacement, and electrical expressions for the thermoelastic piezoelectric contact problem are derived. Using the boundary conditions, the contact problem is reduced to Cauchy-type singular integral equations of the second kind in which the contact stress, the electrical displacement and the contact widths are unknown. The system of singular integral equation is solved numerically using Gauss-Jacobi and Gauss-Chebyshev integration formula. The effect of material inhomogeneity parameter, moving velocity, friction coefficient, external load, electric charge, punch radius on the contact stress, in-plane stress, electric displacement, and generated temperature are discussed in detail.

Free vibration of piezoelectric FGM sandwich beam with porous core embedded in thermal environment and elastic foundation

This paper aims to present thermo-electrical free vibration characteristics of functionally graded material (FGM) sandwich beam placed on the two-parameter elastic foundation. The beam is constructed of a foam core, two middle FGM layers, and two outer piezoelectric layers. It is assumed that the beam is subjected to a constant voltage and a uniform/linear temperature distribution. Physical properties of the core and two middle layers vary smoothly through the thickness according to the cosine and power-law forms, respectively. Lagrange equations in conjunction with the Reddy third-order beam theory is employed to derive the governing equations of motion. A simple polynomial trial function-based Ritz method is adopted for the approximation of the displacement field to obtain the vibration response. The correctness of the study is verified by comparisons with other authors’ published results. Influences of geometry parameters, material property distribution, applied voltage, elastic foundation, temperature distribution, temperature change, porosity coefficient, span-to-height ratio, and boundary conditions are investigated through parametric studies.

Effect of porosity on active vibration control of smart structure using porous functionally graded piezoelectric material

The present work investigates the active vibration control of host structure bonded with functionally graded piezoelectric material (FGPM) as sensor and actuator layer. The effect of porosity volume fraction (ψ) and volume fraction index (λ) on active vibration attenuation is investigated and compared with the dense FGPM sensor and actuator layer. For more real life applications of FGPMs, two piezoelectric materials (PZT-5H and PZT-4) are modelled smoothly in a particular direction. The mechanical and piezoelectric properties of FGPMs vary across the thickness and calculated using power – law expressions. In addition to dense FGPM, two different distributions of the porosities are investigated. Shear deformation theory of first order is used to simulate the vibration response of the host composite plate based on Hamilton’s principle. A well-established negative velocity feedback controller is deployed to obtain active vibration control of the composite structure. The numerical results reveal that the volume fraction index (λ) plays vital role in the active damping of structural vibration. The FGPM-◊ demonstrates 5.7% and 6.32% improved active vibration damping performance compared to FGPM-UD at ψ = 0.1 and ψ = 0.2 respectively. The present investigation can be utilized for further studies on active vibration control of host structures integrated with FGPMs.

Static and dynamic analyses of FGPM cylindrical shells with quadratic thermal gradient distribution

Functionally graded material (FGM) has great potential in the application of thin-walled structures, since delamination failure can be suppressed. Functionally graded piezoelectric material (FGPM) is a special kind of FGM, which couples with elastic and electric fields. The modeling of FGPM coupled smart structures under nonlinear thermal load is still a challenge task. This paper develops an electro-thermo-elastic finite element (FE) model of FGPM integrated smart structures based on the first-order shear deformation hypothesis. The FE model proposes two different nonlinear configurations of thermal gradient distribution through the thickness, which are compared with linear thermal gradient distribution. An eight-node quadrilateral shell element is developed for finite element analysis. The present FE model is firstly validated by a PVDF bimorph beam under nonlinear thermal loads. Later, the model is applied to the static and dynamic simulation of FGPM cylindrical shells. The results show that thermal distribution configurations have great impact on structural response, which should be carefully chosen in the simulation of thermal coupled problems.

Isogeometric optimization of piezoelectric functionally graded material for energy harvester

Improving the energy harvesting efficiency of piezoelectric energy harvesters (PEHs) in the scientific domain is essential for the application of self-powered microelectromechanical devices. This paper proposes an isogeometric optimization method to optimize the material distribution of piezoelectric functionally graded material (PFGM) energy harvester. Firstly, we establish the isogeometric formulation of the PFGM energy harvester based on the plane strain hypothesis and Hamilton’s principle. Meanwhile, the same non-uniform rational B-splines basis functions are utilized to represent the distribution of the PFGM. Secondly, three optimization models with the specified eigenfrequency and maximum energy conversion efficiency as objective functions are established. Then we derive the analytical sensitivities by the adjoint method and use the method of moving asymptotes to update the design variables. Furthermore, several numerical examples are performed to study the convergence and accuracy of the numerical solutions and analyze the effects of the metallic volume ratio constraint and the initial design domain on the optimized material distribution. More importantly, the dynamic analysis is executed to further test the energy harvesting efficiency based on the optimized designs. The results show that the presented approach can not only realize the specified eigenfrequency but also effectively improve the energy harvesting efficiency.

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.