Piezoelectric Nanobeams Research Articles

Effects of porosity and flexoelectricity on static bending and free vibration of AFG piezoelectric nanobeams

Abstract In this paper, based on the strain gradient theory, a size-dependent porous axially functional gradient (AFG) flexoelectric Euler-Bernoulli nanobeam model is developed. A modified power-law formula incorporating porosity volume fraction is presented to describe material properties of porous AFG nanobeam, and two typical porosity distributions patterns are considered. The generalized differential quadrature method is employed to discretize governing equations into a series of linear algebraic equations, so that the static bending deflections and free vibration frequencies are calculated conveniently for various boundary conditions. The credibility of the present mathematical formulation and the associated numerical solution are validated by comparing the results in previous literatures. Parametric studies are performed to illustrate the effects of the flexoelectricity, material length scale parameters, functionally graded index, porosity volume fraction, porosity distribution patterns on the static bending and free vibration behaviors of porous AFG flexoelectric nanobeam. The numerical results may provide support for the application of porous AFG flexoelectric nanobeam in Nano-electro-mechanical system.

Size-dependent static bending and free vibration analysis of porous functionally graded piezoelectric nanobeams

According to electric enthalpy variation and Hamilton’s principle, governing differential equations and boundary conditions of functionally gradient piezoelectric nanobeams with porosities are established. The generalized differential quadrature method plays the role of transforming governing differential equations into a group of linear algebraic equations. Based on the strain gradient theory, a size-dependent functionally gradient piezoelectric nanobeam formulation including additional material length scale parameters is established. The static bending and free vibration analysis of a nanobeam made up of porous functionally gradient piezoelectric materials is researched. Two kinds of porosity distributions are considered in this paper. The influencess of power law index, porosity parameter, porosity distribution, external electrical voltage, flezoelectric effect, length scale parameter and boundary conditions on static deformation and natural frequencies of the nanobeam are researched in detail.

Size-dependent Nonlinear Forced Vibration Analysis of Viscoelastic/Piezoelectric Nano-beam

In this paper, the nonlinear forced vibration of isotropic viscoelastic/ piezoelectric Euler-Bernoulli nano-beam is investigated. For this purpose, the consistent couple stress theory is utilized for modeling the viscoelastic/piezoelectric nano-beam. Hamilton’s principle is also employed to obtain the governing equations of motion. Further, the Galerkin method is used in order to convert the governing partial differential equations to a nonlinear second-order ordinary differential one, and then multiple scale method is used to solve motion equation.

Electro-mechanical coupling wave propagating in a locally resonant piezoelectric/elastic phononic crystal nanobeam with surface effects

The model of a “spring-mass” resonator periodically attached to a piezoelectric/elastic phononic crystal (PC) nanobeam with surface effects is proposed, and the corresponding calculation method of the band structures is formulized and displayed by introducing the Euler beam theory and the surface piezoelectricity theory to the plane wave expansion (PWE) method. In order to reveal the unique wave propagation characteristics of such a model, band structures of locally resonant (LR) elastic PC Euler nanobeams with and without resonators, band structures of LR piezoelectric PC Euler nanobeams with and without resonators, as well as band structures of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on PZT-4, with resonators attached on epoxy, and without resonators are compared. Results demonstrate that adding resonators indeed plays an active role in opening and widening band gaps. Moreover, the influence rules of different parameters on band gaps of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on epoxy are discussed, which will play an active role in the further realization of active control of wave propagations.

Mechanical analysis of cutout piezoelectric nonlocal nanobeam including surface energy effects

This manuscript tends to investigate influences of nanoscale and surface energy on a static bending and free vibration of piezoelectric perforated nanobeam structural element, for the first time. Nonlocal differential elasticity theory of Eringen is manipulated to depict the long–range atoms interactions, by imposing length scale parameter. Surface energy dominated in nanoscale structure, is included in the proposed model by using Gurtin–Murdoch model. The coupling effect between nonlocal elasticity and surface energy is included in the proposed model. Constitutive and governing equations of nonlocal-surface perforated Euler–Bernoulli nanobeam are derived by Hamilton’s principle. The distribution of electric potential for the piezoelectric nanobeam model is assumed to vary as a combination of a cosine and linear variation, which satisfies the Maxwell’s equation. The proposed model is solved numerically by using the finite-element method (FEM). The present model is validated by comparing the obtained results with previously published works. The detailed parametric study is presented to examine effects of the number of holes, perforation size, nonlocal parameter, surface energy, boundary conditions, and external electric voltage on the electro-mechanical behaviors of piezoelectric perforated nanobeams. It is found that the effect of surface stresses becomes more significant as the thickness decreases in the range of nanometers. The effect of number of holes becomes significant in the region 0.2≤ɑ ≤0.8. The current model can be used in design of perforated nano-electro-mechanical systems (PNEMS).

Flexoelectric and surface effects on a cracked piezoelectric nanobeam: Analytical resonant frequency response

Flexoelectric and surface effects on a cracked piezoelectric nanobeam: Analytical resonant frequency response

Flexoelectric effects on wave propagation responses of piezoelectric nanobeams via nonlocal strain gradient higher order beam model

This paper is devoted to investigate the effects of the flexoelectric phenomenon on wave dispersion characteristics of piezoelectric nanobeams. Modified constitutive relations containing the flexoelectric effects are taken into account. The problem is formulated utilizing Reddy’s higher order shear deformation beam theory in conjunction with the nonlocal strain gradient theory (NSGT). Additionally, Hamilton’s principle is employed to derive the governing equations of the problem. Thereafter, considering harmonic wave dispersion, the solution of equations of motion is expressed as complex form. Applying this solution to the governing equations gives rise to the eigenvalue problem. Afterwards, the eigenvalue problem is solved analytically to achieve the wave frequencies and phase velocities of the system. Totally, the role of various involved parameters namely the flexoelectric effects and applied electric voltage on the wave dispersion curves is studied in detail. In addition, the size-dependency of the flexoelectric phenomenon on wave dispersion behavior of the nanobeam is explored. Due to the fact that adequate knowledge about wave dispersion characteristics of structures is of great value especially in nanotechnology field, it is hoped that the results presented here may be helpful for exact behavior determination of piezoelectric nano-structures.

Nonlinear bandgap properties in a nonlocal piezoelectric phononic crystal nanobeam

Applying nonlocal elasticity theory, von Kármán type nonlinear strain-displacement relation and plane wave expansion (PWE) method to Euler-Bernoulli beam, the calculation method of band structure of a nonlinear nonlocal piezoelectric phononic crystal (PC) nanobeam is proposed and formulized. In order to investigate the properties of wave propagating in the nanobeam in detail, band gaps of first four orders are picked, and the corresponding influence rules of electro-mechanical coupling fields, nonlocal effect and geometric parameters on band gaps are studied. During the researches, external electrical voltage and axial force are chosen as the influencing parameters related to electro-mechanical coupling fields. Scale coefficient is chosen as the influencing parameter corresponding to nonlocal effect. Length ratio between materials PZT-4 and epoxy and height-width ratio are chosen as the influencing parameters of geometric parameters. Moreover, all the influence rules are compared to those in linear nanobeam. The results are expected to be of help for the design of micro and nano devices based on piezoelectric periodic nanobeam.

Size-dependent nonlinear bending and vibration of flexoelectric nanobeam based on strain gradient theory

In this study, the nonlinear bending and free vibration of Timoshenko piezoelectric nanobeam incorporating flexoelectricity and surface effect are investigated for the first time. To take size effect into account, a size-dependent Timoshenko nanobeam model containing additional material length scale parameters is developed based on strain gradient theory. The governing equations and corresponding boundary conditions are obtained by electric enthalpy variation and Hamilton’s principle. By adjusting the values of material length scale parameters, the current Timoshenko piezoelectric nanobeam formulations can be transformed to those based on modified couple stress theory. The generalized differential quadrature method (GDQM) is employed to discretize governing differential equations and boundary conditions into a series of nonlinear algebraic equations. Then the algebraic equations are solved by using the Newton iteration method. It is found that strain gradient elastic effect, flexoelectricity, surface effect and applied electric voltage have significant influences on the nonlinear mechanical behaviors of nanobeam. Simulation results indicate that both the strain gradient effect and flexoelectric effect have considerable impacts on the electric field distribution in nanobeams. Moreover, the results also indicate that the influence of flexoelectricity is weakened to some extent due to the consideration of surface effect. The numerical analysis reveals that the present model can be considered reliable to quantitatively investigate size-dependent nonlinear bending and nonlinear free vibration of the piezoelectric nanobeam incorporating flexoelectric and surface effects.

Free vibration of a piezoelectric nanobeam resting on nonlinear Winkler-Pasternak foundation by quadrature methods

This work introduces a numerical scheme for free vibration analysis of elastically supported piezoelectric nanobeam. Based on Hamilton principle, governing equations of the problem are derived. The problem is formulated for linear and nonlinear Winkler–Pasternak foundation type. Three differential quadrature techniques are employed to reduce the problem to an Eigen-value problem. The reduced system is solved iteratively. The natural frequencies of the beam are obtained. Numerical analysis is implemented to investigate computational characteristics affecting convergence, accuracy and efficiency of the proposed scheme. The obtained results agreed with the previous analytical and numerical ones. Furthermore, a parametric study is introduced to show influence of supporting conditions, two different electrical boundary conditions, material characteristics, foundation parameters, temperature change, external electric voltage, nonlocal parameter and beam length-to-thickness ratio on the values of natural frequencies and mode shapes of the problem.

Nonlinear bending and thermal post-buckling behavior of functionally graded piezoelectric nanosize beams using a refined model

We in this paper analyze the problem of the nonlinear bending, thermal buckling and post-buckling of functionally graded piezoelectric material nanobeams based on Eringen’s nonlocal elasticity theory. The beams with immovable clamped ends are exposed to the external electric voltages, a uniform transverse load as well as uniform temperature change. A refined beam model which can transform the Reddy beam model into the Timoshenko beam model is introduced into this study so that we can distinguish the role of transverse shear deformation in complicated stress field. The governing equations are induced by using the generalized variation principle, and then the approximate analytical solution of the FGP material nanobeams for nonlinear, thermal buckling, post-buckling are obtained by using a two-step perturbation method. Subsequently, detailed parametric studies are carried out to get an insight into the effects of different physical parameters, including the slenderness ratio, small scale parameter, volume fraction index and external electric voltages.

Nonlinear free vibration analysis of pre-actuated isotropic piezoelectric cantilever Nano-beams

In this paper, nonlinear free vibration analysis of pre-actuated clamped-free isotropic piezoelectric Euler-Bernoulli nanobeams is discussed. The governing equations of motion are derived on the basis of size-dependent piezoelectricity theory. A more accurate model is developed for the large amplitude vibration analysis of piezoelectric cantliver nanobeams by the consideration of a higher order curvature-displacement relation. In this case, a nonlinear equation of motion is derived. Accordingly, the hardening or softening treatment dependency on the flexoelectric constant and the length scale parameter is examined for the considered nanocantilever. The assumed nano-beam is actuated by a constant voltage. The nonlinear free vibration analysis of pre-actuated nano-beam about the pre-static deformation is examined by Lindstedt-Poincare technique which is applied on the discretized equations of motion. A closed form relation is extracted for the nonlinear natural frequency and the corresponding effective nonlinearity. Some numerical analysis has been performed to peruse the effects of applied voltage, the length scale parameter and flexoelectric coefficient on the static deflection, the nonlinear natural frequencies and the associated effective nonlinearities. The outcomes demonstrate occurrence of very interesting phenomena in the combination of the various magnitudes of the length scale parameter and the flexoelectric coefficient.

Dynamic analysis of a nonlinear nanobeam with flexoelectric actuation

Over the past few years, several researchers have been increasingly attracted to flexoelectric transduction because of its potential application for sensing and actuation in NanoElectroMechanical Systems. The flexoelectric effect refers to coupling between polarization and strain gradient in centrosymmetric and non-centrosymmetric dielectrics. Consequently, not only piezoelectric dielectrics with an initial polarization are of interest, but also a larger range of dielectric materials. In contrast to piezoelectricity, the flexoelectric effect is scale-dependent and can exhibit large electromechanical coefficients only at small scales. This paper focuses on the effects of geometric nonlinearity, resulting from relatively large displacements and restrictive boundary conditions, on the static and dynamic responses of piezoelectric flexoelectric nanobeams. The derived equations of motion for the transverse displacement and variation of the internal electric potential are discretized using a Galerkin procedure. A closed-form solution for the nonlinear static response is proposed. The results are compared and validated with those found in the literature. For the dynamic response, a perturbation technique is used to solve analytically the nonlinear equations of motion for the primary and parametric resonances of the first mode. The analytical perturbation solution is validated using a numerical technique. The results show that a general hardening-type behavior is obtained and, therefore, several jumps are observed for the dynamic solution. High sensitivity of the solution to an applied AC voltage is also demonstrated for the principal resonance of the first mode.

Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

• Extended theory of piezoelectricity is used to investigate flexoelectricity effect on vibrational behavior of nanobeam. • The von-Kármán strain–displacement relationship, electric Gibbs free energy and Hamilton's principle are employed. • Multiple scales method is used to obtain a closed form solution of derived nonlinear governing equations of motion. • The effects of changes in parameters on free vibration and postbuckling behavior of nanobeam are discussed comprehensively. • Theoretical range of flexoelectricity constant is estimated through buckling analysis . This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.

Size-Dependent Buckling and Vibrations of Piezoelectric Nanobeam with Finite Element Method

In the present paper, a finite element method is used to study the vibrations and buckling of a piezoelectric nanobeam. The beam theory used here is Bernoulli–Euler model. In order to achieve the goal, first, the governing equations of the piezoelectric nanobeam were obtained via the Hamilton principle, based on the higher-order theory such as modified couple stress. The shape functions were embedded in these equations, and the matrix form of the equations was obtained. In other words, the governing equations were discretized by the finite element method. By calculating the matrix form of equations, mass matrices, mechanical stiffness and electrical stiffness of the beam were obtained. It should be noted that when calculating stiffness matrices, first, the definition of a new element must be added to the effects of the electric field in the elements of the beam, and the effects of the gradient of strain should be added in the form of non-classical matrix to the stiffness matrix. With the aid of these critical load matrices, the buckling force and voltage and the natural frequency of the piezoelectric nanobeam have been calculated. Numerical results show that this method can involve the size effects.

Dynamic instability region analysis of sandwich piezoelectric nano-beam with FG-CNTRCs face-sheets based on various high-order shear deformation and nonlocal strain gradient theory

In this research, the dynamic instability region (DIR) of the sandwich nano-beams are investigated based on nonlocal strain gradient elasticity theory (NSGET) and various higher order shear deformation beam theories (HSDBTs). The sandwich piezoelectric nano-beam is including a homogenous core and face-sheets reinforced with functionally graded (FG) carbon nanotubes (CNTs). In present study, three patterns of CNTs are employed in order to reinforce the top and bottom face-sheets of the beam. In addition, different higher-order shear deformation beam theories such as trigonometric shear deformation beam theory (TSDBT), exponential shear deformation beam theory (ESDBT), hyperbolic shear deformation beam theory (HSDBT), and Aydogdu shear deformation beam theory (ASDBT) are considered to extract the governing equations for different boundary conditions. The beam is subjected to thermal and electrical loads while is resting on Visco-Pasternak foundation. Hamilton principle is used to derive the governing equations of motion based on various shear deformation theories. In order to analysis of the dynamic instability behaviors, the linear governing equations of motion are solved using differential quadrature method (DQM). After verification with validated reference, comprehensive numerical results are presented to investigate the influence of important parameters such as various shear deformation theories, nonlocal parameter, strain gradient parameter, the volume fraction of the CNTs, various distributions of the CNTs, different boundary conditions, dimensionless geometric parameters, Visco-Pasternak foundation parameters, applied voltage and temperature change on the dynamic instability characteristics of sandwich piezoelectric nano-beam.

Dynamic instability region analysis of sandwich piezoelectric nano-beam with FG-CNTRCs face-sheets based on various high-order shear deformation and nonlocal strain gradient theory

In this research, the dynamic instability region (DIR) of the sandwich nano-beams are investigated based on nonlocal strain gradient elasticity theory (NSGET) and various higher order shear deformation beam theories (HSDBTs). The sandwich piezoelectric nano-beam is including a homogenous core and face-sheets reinforced with functionally graded (FG) carbon nanotubes (CNTs). In present study, three patterns of CNTs are employed in order to reinforce the top and bottom face-sheets of the beam. In addition, different higher-order shear deformation beam theories such as trigonometric shear deformation beam theory (TSDBT), exponential shear deformation beam theory (ESDBT), hyperbolic shear deformation beam theory (HSDBT), and Aydogdu shear deformation beam theory (ASDBT) are considered to extract the governing equations for different boundary conditions. The beam is subjected to thermal and electrical loads while is resting on Visco-Pasternak foundation. Hamilton principle is used to derive the governing equations of motion based on various shear deformation theories. In order to analysis of the dynamic instability behaviors, the linear governing equations of motion are solved using differential quadrature method (DQM). After verification with validated reference, comprehensive numerical results are presented to investigate the influence of important parameters such as various shear deformation theories, nonlocal parameter, strain gradient parameter, the volume fraction of the CNTs, various distributions of the CNTs, different boundary conditions, dimensionless geometric parameters, Visco-Pasternak foundation parameters, applied voltage and temperature change on the dynamic instability characteristics of sandwich piezoelectric nano-beam.

Bandgap properties of a piezoelectric phononic crystal nanobeam based on nonlocal theory

The aim of this paper is to investigate the bandgap properties of a piezoelectric phononic crystal (PC) nanobeam with size effect by coupling the plane wave expansion method, Euler–Bernoulli beam theory and nonlocal theory. The first four orders were chosen to study the influences of thermo-electro coupling, size effect and geometric parameters on band gaps. Temperature change and external electrical voltage were chosen as the parameters capable of influencing thermo-electro coupling fields. Scale coefficient was chosen as the influencing parameters related to size effect. The lengths of PZT-4 and epoxy within a unit cell, along with the width and thickness of the PC nanobeam, were identified as influential geometric parameters. Collectively, our results are expected to be helpful for the design of piezoelectric nanobeam-based devices.

Influence of Flexoelectricity on Electromechanical Properties of Functionally Graded Piezoelectric Nanobeams Based on Modified Couple Stress Theory

Flexoelectricity refers to the linear coupling between the strain gradient and the electric polarization at micro/nanometer scale and exists in all dielectrics. Flexoelectricity exhibits strong size-dependence, which can be remarkably enhanced by size reduction. Flexoelectricity is also shape-associated and the strain gradients induced electric polarization in nonpiezoelectric dielectrics with irregular shapes (i.e., truncated pyramids) depend on shape geometry, dimension and aspect ratio of the structures. In this work, a novel asymmetric piezoelectric nanobeam based on functionally graded (FG) nanomaterial is modeled and the flexoelectricity in such a designed structure is theoretically examined. Since the material properties of FG nanomaterial vary along a certain direction, it is expected that such an inhomogeneity will result in large strain gradients in the nanobeam and a large flexoelectric response will occur. By assuming that the material properties of FG nanomaterial obey a power volume fraction distribution, we derive a six-order governing equation based on modified couple stress elasticity theory, and closed-form solutions are obtained. Results indicate that the flexoelectricity has a significant influence on the electromechanical responses in FG piezoelectric nanobeams. The results obtained from the current work may help explain the extremely large flexoelectric coefficients observed in some classes of materials with irregular shapes.

On the size-dependent magneto/electromechanical buckling of nanobeams

Bending and buckling of piezoelectric and piezomagnetic nanobeams are investigated based upon the Euler-Bernoulli beam model and using the strain gradient theory, which is capable of accounting for higher-order magneto/electromechanical coupling as well as size effects. Governing equations and boundary conditions are extracted using the principle of minimum potential energy, and electrical and mechanical field distributions are computed using the equations obtained. The size effect is accounted for using the strain gradient theory, and nanobeam buckling is analyzed using the nonlinear von Karman strain. Nanobeam bending and buckling are examined through the Galerkin procedure. The impact of parameters such as size effects, length, thickness, material properties, external voltage, and magnetic potential is investigated, and critical load, critical voltage, and critical magnetic potential of buckling are shown to be considerably dependent upon size effects, particularly as thickness increases.