Piezoelectric Nanoplates Research Articles

Wave Propagation Analysis of Piezoelectric Nanoplates Based on the Nonlocal Theory

Based on the nonlocal theory, this paper develops the Kirchhoff nanoplate and Mindlin nanoplate models for the wave propagation analysis of piezoelectric nanoplates. The effects of small scale parameter and thermo-electro-mechanical loads are incorporated in the nanoplate models. The Hamilton’s principle is employed to derive the governing equations of the nanoplate, which are solved analytically to obtain the dispersion relation for piezoelectric nanoplates. The results show that the nonlocal parameter, temperature change, mechanical load and external electric potential have significant influence on the wave propagation characteristics of the piezoelectric nanoplates. The cut-off wave number is observed to exist for piezoelectric nanoplates subjected to positive electric potential, axial tensile force and temperature rise.

Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory

The effective elastic-piezoelectric properties of nanostructures have been shown to be strongly size-dependent. In this paper, a nonlocal second-order shear deformation formulation is presented to study the size-dependent thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core. Temperature is considered as uniform and nonlinear distributions across plate’s thickness direction. Based on the developed nonlocal second-order shear deformation theory, the size-dependent equations of motion are derived. The nonlocal thermal buckling responses of simply supported nanoplates are solved via Navier method. The reliability of present approach is verified by comparing the existing results provided in the open literature. The influences of nonlocal parameter, gradient index, electric voltage, and Winkler–Pasternak parameters on the thermal buckling characteristics of functionally graded nanoplates are examined.

Fundamental frequency analysis of rectangular piezoelectric nanoplate under in‐plane forces based on surface layer, non‐local elasticity, and two variable refined plate hypotheses

Fundamental frequency analysis of rectangular piezoelectric nanoplates via the surface layer and non-local small-scale hypotheses is investigated in the present work. The piezoelectric nanoplate is under in-plane forces. The equilibrium governing of piezoelectric nanoplates is attained via the two variable refined plate hypothesis, and then the equations of motion are achieved utilising Hamilton's principle. To solve these equations, the finite difference method is employed. To verify the exactness of the finite difference method, the governing equations are tested by the Navier's solution. Numerical results show a good accuracy among the outcomes of the present work and some accessible cases in the literature. The numerical results show that for negative residual surface stress, as the boundary condition becomes stiffer the effect of surface layer increases, while for positive one that phenomenon is inverse.

Temperature influences on shear stability of a nanosize plate with piezoelectricity effect

PurposeThe purpose of this paper is to predict the mechanical behavior of a piezoelectric nanoplate under shear stability by taking electric voltage into account in thermal environment.Design/methodology/approachSimplified first-order shear deformation theory has been used as a displacement field. Modified couple stress theory has been applied for considering small-size effects. An analytical solution has been taken into account for various boundary conditions.FindingsThe length scale impact on the results of any boundary conditions increases with an increase in l parameter. The effect of external electric voltage on the critical shear load is more than room temperature effects. With increasing aspect ratio the critical shear load decreases and external electric voltage becomes more impressive. By considering piezoelectric nanoplates, it is proved that the temperature rise cannot become a sensitive factor on the buckling behavior. The length scale parameter has more effect for more flexible boundary conditions than others. By considering nanosize, the consideration has led to much bigger critical load vs macro plate.Originality/valueIn the current paper for the first time the simplified first-order shear deformation theory is used for obtaining governing equations by using nonlinear strains for shear buckling of a piezoelectric nanoplate. The couple stress theory for the first time is applied on the nonlinear first-order shear deformation theory. For the first time, the thermal environment effects are considered on shear stability of a piezoelectric nanoplate.

Coupled effects of electrical polarization-strain gradient on vibration behavior of double-layered flexoelectric nanoplates

A vibrating double-layered nanoscale piezoelectric plate is developed accounting for the flexoelectricity and surface effects. The flexoelectricity is due to the coupling between electrical polarization and strain gradient. Applying Hamilton\'s principle, the governing equations and related boundary conditions are derived. Assuming suitable approximate functions, the governing equations are numerically solved for simply-supported and clamped boundary conditions. Obtained results indicate that both the flexoelectricity and surface effects possess notable impact on the vibration frequencies of the system. Only flexoelectricity yields a considerable difference between the present model and previous investigations on conventional piezoelectric nanoplates. Generally, a parametric study has been performed to examine the effects of surface elasticity, flexoelectricity, applied electric voltage, interlayer stiffness, geometrical parameters and boundary conditions on vibration frequencies of piezoelectric nanoplates.

Wave propagation analysis of embedded nanoplates based on a nonlocal strain gradient-based surface piezoelectricity theory

The present paper deals with the smart characteristics of waves propagating in a piezoelectric nanosize plate rested on an elastic medium whenever surface effects are included. A more realistic simulation about the elastic medium is presented by utilizing a three-parameter medium containing Winkler, Pasternak and damping coefficients. Furthermore, both of the decreasing and increasing impacts of small scale influences are covered in the framework of a nonlocal strain gradient theory (NSGT). The electric potential is approximated by a function possessing linear and trigonometric parts. Also, by developing the surface elasticity theory of Gurtin-Murdoch for piezoelectric solids, influences of surface layers are considered. Kinematic relations are derived employing the Kirchhoff plate theory. Afterwards, Hamilton’s principle is introduced in order to achieve Euler-Lagrange equations of piezoelectric nanoplates. The final part consists of an analytical approach to obtain the wave frequency value. The accuracy of the presented model is verified by organizing a comparison of the presented results with previous ones. Finally, some parametric case studies are rendered to clarify the influence of different parameters such as wave number, nonlocal and length scale parameters, foundation parameters and applied voltage.

Bending deformation of multilayered one-dimensional hexagonal piezoelectric quasicrystal nanoplates with nonlocal effect

Based on the nonlocal elasticity theory, the static bending deformation of one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) nanoplates is investigated under surface electroelastic loadings. The general solutions for the extended displacement and traction vectors of a simply supported and homogeneous PQC nanoplate are derived by solving an eigenvalue problem reduced from the pseudo-Stroh formalism. By utilizing the propagator matrix method, exact closed-form solutions of multilayered 1D hexagonal PQC nanoplates are then obtained by assuming that the layer interfaces are perfectly contacted. Numerical examples for six kinds of sandwich nanoplates made up of piezoelectric crystals (PE), quasicrystal (QC) and PQC are presented to illustrate the effect of the nonlocal parameter and stacking sequence of the nanoplates on the phonon, phason and electric fields, which play an important role in designing new composite structures in engineering.

Static stability analysis of embedded flexoelectric nanoplates considering surface effects

In this paper, electromechanical buckling behavior of size-dependent flexoelectric nanoplates is investigated based on nonlocal and surface elasticity theories. Flexoelectricity represents the coupling between strain gradients and electrical polarizations. Flexoelectric nanoplates can tolerate higher buckling loads compared with conventional piezoelectric nanoplates, especially at lower thicknesses. The flexoelectric nanoplate is in contact with a two-parameter elastic foundation, which consists of infinite linear springs and a shear layer. Nonlocal elasticity theory of Eringen is applied in the analysis of flexoelectric nanoplates for the first time. The residual surface stresses which are usually neglected in the modeling of flexoelectric nanoplates are incorporated into nonlocal elasticity to provide better understanding of the physics of the problem. Applying an analytical solution which satisfies various boundary conditions, the governing equations obtained from Hamilton’s principle are solved. The reliability of the present approach is verified by comparing the obtained results with those provided in literature. Finally, the influences of nonlocal parameter, surface effect, plate geometrical parameters, elastic foundation and boundary conditions on the buckling characteristics of flexoelectric nanoplates are explored.

Nonlocal piezo-hygrothermal analysis for vibration characteristics of a piezoelectric Kelvin–Voigt viscoelastic nanoplate embedded in a viscoelastic medium

This article presents vibration characteristics of a piezoelectric viscoelastic nanoplate under various boundary conditions embedded in a visco-Pasternak’s medium using nonlocal plate theory. The piezo-hygrothermal effects are all included. The hygrothermal piezoelectric nanoplate is made of orthotropic viscoelastic material. Also, the visco-Pasternak’s medium is modeled according to a Kelvin–Voigt foundation. A Two-variable sinusoidal shear deformation plate theory is presented. The imaginary and real parts of the eigenfrequency of piezoelectric viscoelastic nanoplates which are associated with viscoelastic foundation, nonlocal parameter, as well as the viscoelastic structural damping coefficient, are obtained. The effects of other parameters such as aspect ratio, side-to-thickness ratio, temperature change, and moisture concentration are all investigated. The predicted results are compared with the corresponding ones in the literature to check their validation. Additional results are presented and suitable discussions are made.

Thermo-electro-mechanical vibration analysis of piezoelectric nanoplates resting on viscoelastic foundation with various boundary conditions

Based on the nonlocal Kirchhoff plate theory, the thermo-electro-mechanical vibration responses are studied for a rectangular piezoelectric nanoplate resting on viscoelastic foundation with various boundary conditions. In doing this, the governing equations of motion are first derived for vibration analysis, where thermo-electro-mechanical loadings, nonlocal effect, piezoelectric effect and viscoelastic foundation have been taken into consideration. Subsequently, the Galerkin strip distributed transfer function method is proposed to solve the governing equations, which enables one to obtain the semi-analytical solutions of natural frequencies for piezoelectric nanoplates with arbitrary boundary conditions. Here, the developed mechanics model is first validated by comparing the obtained results with those available in the literature. Also, the effects of nonlocal parameter, boundary conditions, viscoelastic foundation, external electric voltage, increment temperature, biaxial force and geometric dimensions on the vibration behaviors are carefully examined for the piezoelectric nanoplate. The results demonstrate the efficiency and robustness of the developed model for vibration analysis of a complicated multi-physics system comprising piezoelectric nanoplates, viscoelastic foundation and thermo-electro-mechanical loadings.

Buckling Analysis of Double-Layer Piezoelectric Nanoplates Surrounded by Elastic Foundations and Thermal Environments Considering Nonlocal and Surface Energy Models

AbstractThis study investigated the effects of considering surface and nonlocal energy parameters on the buckling analysis of double piezoelectric nanoplate (DPNP) embedded in elastic foundations and thermal environments. Both in-phase and out-of-phase modes of buckling and various boundary conditions are studied and compared with each other. The governing equations were derived by drawing on the principle of virtual work and then solved by employing the finite difference method. Finite difference solution was validated using Navier's method and journal references. A parametric study was also launched in order to investigate the effects of the external electric voltage, nonlocal parameters, different boundary conditions, elastic foundations and thermal environments on the surface effect of DPNP buckling. The obtained numerical results showed that the influence of surface stress on in-phase and out-of-phase modes of buckling of the DPNP was enhanced by augmenting the nonlocal parameters and external electric voltage; on the other hand, it was found to be decreased by increasing elastic foundations and temperature changes. In addition, the value of surface stress effects for the in-phase mode was higher than that of the out-of-phase one.

The effects of surface stress and nonlocal small scale on the uniaxial and biaxial buckling of the rectangular piezoelectric nanoplate based on the two variable-refined plate theory

The effects of surface stress and nonlocal small scale on the uniaxial and biaxial buckling of the rectangular and skew piezoelectric nanoplate were investigated based on the two variable-refined plate theory (TVRPT). The governing equations were derived using the principle of virtual work and then it was solved using the finite difference method. Finite difference solutions were validated by Navier’s method and journal references. A parametric study was conducted to investigate the influences of the external electric voltage, nonlocal parameter, different boundary condition, nanoplate thickness, and aspect ratio on the surface effect of uniaxial and biaxial buckling. Numerical results obtained showed that the influence of surface stress on the buckling of the rectangular piezoelectric nanoplate was enhanced by augmenting the nonlocal parameter, external electric voltage, and aspect ratio while it was decreased in the thicker nanoplate. The present work could be very useful in designing micro and nano-devices using smart nanocomposites.

A Study of the Flexoelectric Effect on the Electroelastic Fields of a Cantilevered Piezoelectric Nanoplate

Flexoelectricity, a spontaneous polarization in linear response to strain gradients or non-uniform deformation, is believed to contribute to the size-dependent electromechanical coupling of piezoelectric materials at the nanoscale. In the current work, the flexoelectric effect upon the static bending behaviors of a cantilevered piezoelectric nanoplate (PNP) is studied. Based on the Kirchhoff plate model and the extended linear piezoelectric theory, the non-conventional governing equations and the boundary conditions of the PNP under both mechanical and electrical loads are derived with the incorporation of the flexoelectric effect. Finite difference method (FDM) is performed to get the numerical solution for the electroelastic fields of the plate. Simulation results show that the flexoelectric effect is more prominent for the thinner plates with smaller thickness. It is also found that the flexoelectric effect upon the electroelastic responses of the clamped PNP is also sensitive to some other factors, including the boundary conditions, the plate geometric ratio, and the applied mechanical and electrical loads. This work aims to provide an increased understanding of the size-dependent electromechanical coupling properties of a piezoelectric plate structure.

Damping vibration analysis of smart piezoelectric polymeric nanoplates on viscoelastic substrate based on nonlocal strain gradient theory

This paper develops a nonlocal strain gradient plate model for damping vibration analysis of smart piezoelectric polymeric nanoplates resting on visco-Pasternak medium. For more accurate analysis of piezoelectric nanoplate, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Viscoelastic effect is considered based on Kelvin–Voit model. Governing equations of a nonlocal strain gradient smart nanoplate on viscoelastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations. Effects of different factors such as viscoelasticity, nonlocal parameter, length scale parameter, applied voltage and Winkler–Pasternak parameters on damping vibration characteristics of a nanoplate are studied.

Size-dependent free vibration analysis of functionally graded piezoelectric plate subjected to thermo-electro-mechanical loading

Analytical solutions are presented to study the thermo-electro-mechanical vibration problem of functionally graded piezoelectric materials’ nanoscale rectangular plates based on the size-dependent Kirchhoff plate theory. The plate is under an applied electric voltage, a biaxial force, and a uniform temperature rise. The material properties vary continuously along the thickness direction. To check the validity of the method, the results are compared with the results obtained in previous studies for the special cases. Finally, a parametric study is accompanied to survey the effects of the gradient index, length scale parameter, length-to-thickness ratio, external electric, and temperature change on the vibration of functionally graded piezoelectric nanoplates.

Vibration analysis of size-dependent flexoelectric nanoplates incorporating surface and thermal effects

ABSTRACTThis article is concerned with the thermo-mechanical vibration behavior of flexoelectric nanoplates under uniform and linear temperature distributions. Flexoelectric nanoplates have higher natural frequencies than conventional piezoelectric nanoplates, especially at lower thicknesses. Both nonlocal and surface effects are considered in the analysis of flexoelectric nanoplates for the first time. Hamilton's principle is employed to derive the governing equations and the related boundary conditions, which are solved applying the Galerkin-based solution. A comparison study is also performed to verify the present formulation with those of previous data. Numerical results are presented to investigate the influences of the flexoelectricity, nonlocal parameters, surface elasticity, temperature rise, plate thickness, and various boundary conditions on the vibration frequencies of thermally affected flexoelectric nanoplate.

Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory

• Shear buckling of piezoelectric nanoplate is analyzed based on modified couple stress theory. • Nonlinear equations are found using simplified first order shear deformation theory. • The critical shear load is obtained via closed-form solution with various boundary conditions. • Increasing length scale parameter increases critical shear load. • The effect of external electric voltage on the critical shear load occurring on the piezoelectric nanoplate is insignificant. This paper studies the electro-mechanical shear buckling analysis of piezoelectric nanoplate using modified couple stress theory with various boundary conditions.In order to be taken electric effects into account, an external electric voltage is applied on the piezoelectric nanoplate. The simplified first order shear deformation theory (S-FSDT) has been employed and the governing differential equations have been obtained using Hamilton's principle and nonlinear strains of Von-Karman. The modified couple stress theory has been applied to considering small scale effects. An analytical approach was developing to obtain exact results with various boundary conditions. After all, results have been presented by change in some parameters, such as; aspect ratio, effect of various boundary conditions, electric voltage and length scale parameter influences. At the end, results showed that the effect of external electric voltage on the critical shear load occurring on the piezoelectric nanoplate is insignificant.

Surface layer and nonlocal parameter effects on the in-phase and out-of-phase natural frequencies of a double-layer piezoelectric nanoplate under thermo-electro-mechanical loadings

The in-phase and out-of-phase natural frequencies of double-layer piezoelectric nanoplates (DPNP) are studied under thermo-electro-mechanical loadings using the surface layer and nonlocal elasticity theories. For this purpose, two nanoplates are coupled via an elastic medium that induces van der Waals forces while the DPNP is surrounded by the Winkler and shear moduli of elastic foundations. Both in-phase and out-of-phase fundamental frequencies are investigated and compared under different boundary conditions. The Hamilton’s principle is employed to derive the governing equations and the finite difference method is subsequently used to solve them. The finite difference solution thus obtained is validated against the results obtained from the Navier’s method. A different aspect of the study involves the investigation of the effects of nonlocal parameters, thermo-electro-mechanical forces, various boundary conditions, and elastic foundations on the surface layer vibration of a DPNP. It is observed that the effects of surface energy layers on the in-phase and out-of-phase vibrations increase with increasing values of the nonlocal parameter and external electric voltage but that they reduce with rising mechanical and thermal loadings, elastic foundation, and stiffness for the boundary conditions defined. The results of the present work are expected to be used in designing NEMS/MEMS components using smart composite nanostructures.

The nonlocal and gradient theories for a large deformation of piezoelectric nanoplates

The von Karman large deformations are considered in the Mindlin plate theory described by the nonlocal and gradient elasticity for piezoelectric nanoplates. It is shown that electric intensity vector can be expressed by mechanical quantities. The governing equations for bending moments, normal and shear stresses are derived from the variational principle. The finite element method is developed for considered governing equations. Differences of both theories are presented.

Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces

Abstract This work is motivated by the self-powered component of biomedical nano-robotic device which is expected to move in arterial blood vessels. The transverse vibrations and steady-state responses of axially moving viscoelastic piezoelectric two-dimensional nanostructures are investigated based on the nonlocal viscoelasticity thin plate theory. The constitutive relations of viscoelastic piezoelectric nanoplate containing the thermal effect are performed and the governing partial differential equations of the problem model are derived using the Hamilton's principle. The natural frequencies are numerically determined via the Galerkin method, the complex mode method, as well as the finite element method for comparison. Moreover, the theoretical calculations are compared with those in previous literature to verify effects in nonlocal nanoscale framework and average speed on natural frequency. Afterwards, the instable behaviors of axially non-uniformly moving viscoelastic piezoelectric nanoplate characterized as a sine variation about the constant average speed are addressed using the method of multiple scales. The analyses are mainly focused on the boundaries of instable regions in combination parametric resonance and principal parametric resonance. The non-dimensional numerical results imply the existences of nonlocal nanoscale parameter and average speed contribute to reduce the rigidity, and further produce the coupled vibrations and flutter instabilities for complex frequencies. Additionally, the instable regions of combination and principal parametric resonances decrease with increases in the biaxial compression, change of temperature, positive electric voltage and viscoelastic coefficient. The dynamic responses of axially moving viscoelastic piezoelectric nanoplate under the coupling of thermo-electro-mechanical multi-fields are expected to play significant roles in designing biomedical nano-robots.