Piezoelectric Nanobeams Research Articles

Investigation of the Size Effect on the Nano-beam Type Piezoelectric Low Power Energy Harvesting

In this paper, size dependent beam type peizoelectric energy hardvester is investigated. For this goal, first nonlinear formulation of isotropic piezoelectric Euler-Bernoulli nano-beam is developed based on the size-dependent piezoelectricity theory then special beam type piezoelectric energy hardvester is probed for different parameters. Basic nonlinear equations of piezoelectric nano-beam are derived using principle of minimum of potential energy and variational method. To evaluate the formulation derived, static deformation and free vibration of the clamped-clamped piezoelectric nano-beam is investigated in the special case. The results of the formulation derived are investigated under different parameters, and particularly, the ability and performance of the beam type piezoelectric low power energy harvesting was evaluated in nanoscale.

Bandgap properties of a piezoelectric phononic crystal nanobeam with surface effect

The aim of this paper was to extend the traditional plane wave expansion method to calculate the band structure of a proposed piezoelectric phononic crystal (PC) nanobeam with surface effect. The first four orders were chosen to investigate the influence of electrical fields, surface effects, and geometric parameters upon bandgaps. We chose the external electrical voltage applied as the influencing parameter related to electric field. Residual surface stress and material intrinsic length were chosen as the influencing parameters related to surface effect. The influencing geometric parameters were the length of PZT-5H material and epoxy in a unit cell, along with the width and thickness of the PC nanobeam. We hypothesized that data arising from these investigations would be helpful in designing piezoelectric nanobeam-based devices.

Effect of surface stress and surface-induced stress on behavior of piezoelectric nanobeam

A new continuum model is developed to study the influence of surface stress on the behaviors of piezoelectric nanobeams. Different from existing piezoelectric surface models which only consider the surface properties, the proposed model takes surfaceinduced initial fields into consideration. Due to the fact that the surface-induced initial fields are totally different under various boundary conditions, two kinds of beams, the doubly-clamped beam and the cantilever beam, are analyzed. Furthermore, boundary conditions can affect not only the initial state of the piezoelectric nanobeam but also the forms of the governing equations. Based on the Euler-Bernoulli beam theory, the nonlinear Green-Lagrangian strain-displacement relationship is applied. In addition, the surface area change is also considered in the proposed model. The governing equations of the doubly-clamped and cantilever beams are derived by the energy variation principle. Compared with existing Young-Laplace models, the proposed model for the doubly-clamped beam is similar to the Young-Laplace models. However, the governing equation of the cantilever beam derived by the proposed model is very different from that derived by the Young-Laplace models. The behaviors of piezoelectric nanobeams predicted by these two models also have significant discrepancies, which is owing to the surface-induced initial fields in the bulk beam.

Buckling of a piezoelectric nanobeam with interfacial imperfection and van der Waals force: Is nonlocal effect really always dominant?

With the development of NEMSs (nano-electro-mechanical systems), the size-dependent behavior has been an active topic. The scale parameter seems an essential factor dominating mechanical behavior at nanoscale. However, is it always dominant? This is a question deserving careful investigation. For this reason, the buckling of a bi-layered PE (piezoelectric) nanobeam is analyzed. The main purpose is to reveal and compare the effects of the nonlocal scale, imperfect interface, interlaminar van der Waals force and loading ratio. The imperfect interface is modeled by normal and shear springs, and the van der Waals force is represented by the Hamaker formula. Based on the principle of virtual work, the analytical solution of the critical buckling loading is derived by using the trigonometric shear deformation theory. After the verification in some degenerated cases, parametric studies are conducted. It is indicated that if the nonlocal parameter varies in the typical range [0, 4 nm^2], it only has quite limited effect on the buckling behavior, as compared with the other three factors. In this case, although the buckling relies on the nonlocal scale, it is far more dependent on the conventional non-nanoscale factors. The conclusions can provide references for optimal design of NEMSs.

Comprehensive nonlocal analysis of piezoelectric nanobeams with surface effects in bending, buckling and vibrations under magneto-electro-thermo-mechanical loading

The bending, buckling and vibrational behavior of size-dependent piezoelectric nanobeams under thermo-magneto-mechano-electrical environment are investigated by performing a parametric study, in the presence of surface effects. The Gurtin-Murdoch surface elasticity and Eringen’s nonlocal elasticity theories are applied in the framework of Euler–Bernoulli beam theory to obtain a new non-classical size-dependent beam model for dynamic and static analyses of piezoelectric nanobeams. In order to satisfy the surface equilibrium equations, cubic variation of stress with beam thickness is assumed for the bulk stress component which is neglected in classical beam models. Results are obtained for clamped - simply-supported (C-S) and simply-supported - simply-supported (S-S) boundary conditions using a proposed analytical solution method. Numerical examples are presented to demonstrate the effects of length, surface effects, nonlocal parameter and environmental changes (temperature, magnetic field and external voltage) on deflection, critical buckling load and natural frequency for each boundary condition. Results of this study can serve as benchmarks for the design and analysis of nanostructures of magneto-electro-thermo-elastic materials.

Thermal post-buckling analysis of closed circuit flexoelectric nanobeams with surface effects and geometrical imperfection

ABSTRACTThermal post-buckling analysis of a geometrically imperfect nanoscale piezoelectric beam under closed circuit condition is performed 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. The post-buckling load-deflection relation is obtained by solving the governing equations having cubic nonlinearity applying Galerkin's method needless of any iteration process. Obtained results indicate that both the flexoelectricity and surface effects possess notable impact on the post-buckling behavior of the system. Only flexoelectricity yields a considerable difference between the present model and previous investigations on conventional piezoelectric nanobeams. A parametric study has been performed to examine the effects of surface elasticity, flexoelectricity, applied electric voltage, geometrical imperfection, foundation parameters and boundary conditions on post-buckling load of piezoelectric nanobeams.

Theoretical and finite element modeling of piezoelectric nanobeams with surface and flexoelectricity effects

ABSTRACTIn this paper, a size-dependent Euler–Bernoulli beam model, which takes the flexoelectricity, piezoelectricity, and dielectricity as well as the surface elasticity into consideration, is established. Theoretical solutions for the static bending deflection of thin beams under different loading (uniformly distributed and concentrated load) and boundary conditions (cantilevered, both ends simply supported, clamped–clamped), are established. Moreover, an iterative finite element algorithm is developed for the analysis of thin flexoelectric beams under any load and boundary conditions. Our numerical results show the direct bulk flexoelectricity always play the role of stiffening the beams with various boundary conditions, while the residual surface stresses behave either stiffening or softening the nanobeams dependent on boundary conditions. The present investigation also demonstrates the size-dependent effects of flexoelectricity on the bending rigidity and piezoelectricity.

Flexoelectric effect on vibration responses of piezoelectric nanobeams embedded in viscoelastic medium based on nonlocal elasticity theory

In this study, vibration characteristics of a piezoelectric nanobeam embedded in a viscoelastic medium are investigated based on nonlocal Euler–Bernoulli beam theory. In doing this, the governing equations of motion and boundary conditions for vibration analysis are first derived using Hamilton’s principle, where nonlocal effect, piezoelectric effect, flexoelectric effect, and viscoelastic medium are considered simultaneously. Subsequently, the transfer function method is employed to obtain the natural frequencies and corresponding mode shapes in closed form for the embedded piezoelectric nanobeam with arbitrary boundary conditions. The proposed mechanics model is validated by comparing the obtained results with those available in the literature, where good agreement is achieved. The effects of nonlocal parameter, boundary conditions, slenderness ratio, flexoelectric coefficient, and viscoelastic medium on vibration responses are also examined carefully for the embedded nanobeam. The results demonstrate the efficiency and robustness of the developed model for vibration analysis of a complicated multi-physics system comprising piezoelectric nanobeam with flexoelectric effect, viscoelastic medium, and electrical loadings.

Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity

ABSTRACTThis manuscript illustrates coupled effects of nonlocal elasticity and surface properties on static and vibration characteristics of piezoelectric nanobeams using thin beam theory. The mechanical and piezoelectric surface nanoscale properties are governed by Gurtin–Murdoch model. The length scale effect is imposed to the problem by including nonlocal elasticity theory to describe the long–range atoms interactions. The governing equations are derived by Hamilton’s principle and solved numerically using the finite-element method. The proposed model is verified and validated with previous published works. Numerical results illustrate the effects of nonlocal parameter, surface elasticity, and boundary conditions on the bending and dynamic characteristics of the nanobeam. It is found that, the nonlocal effect and the surface piezoelectricity effect play a significant role on the static deflection and the natural frequencies. The obtained results are in good agreement with previous published works. This study should have useful insights on the design, fabrication and applications of piezoelectric-beam-based nanodevices.

Nonlocal and surface effects on the buckling behavior of flexoelectric sandwich nanobeams

ABSTRACTIn this article, an analytical approach is presented to study the surface and flexoelectric effects on the buckling characteristics of an embedded piezoelectric sandwich nanobeam. According to the nonlocal elasticity theory, the flexoelectricity is believed to be authentic for size-dependent properties in nanostructures. The boundary conditions and the governing equations are derived by Hamilton's principle and are solved by Navier method. The results obtained from the present work show that the nonlocal term has an important reduction on the critical load and also the flexoelectricity shows an increasing influence on the buckling loads of the sandwich nanobeam, especially at lower thicknesses.

Analytical wave dispersion modeling in advanced piezoelectric double-layered nanobeam systems

This research deals with the wave dispersion analysis of functionally graded double-layered nanobeam systems (FG-DNBSs) considering the piezoelectric effect based on nonlocal strain gradient theory. The nanobeam is modeled via Euler-Bernoulli beam theory. Material properties are considered to change gradually along the nanobeams\' thickness on the basis of the rule of mixture. By implementing a Hamiltonian approach, the Euler-Lagrange equations of piezoelectric FG-DNBSs are obtained. Furthermore, applying an analytical solution, the dispersion relations of smart FG-DNBSs are derived by solving an eigenvalue problem. The effects of various parameters such as nonlocality, length scale parameter, interlayer stiffness, applied electric voltage, relative motions and gradient index on the wave dispersion characteristics of nanoscale beam have been investigated. Also, validity of reported results is proven in the framework of a diagram showing the convergence of this model\'s curve with that of a previous published attempt.

Magnetic field effects on buckling characteristics of smart flexoelectrically actuated piezoelectric nanobeams based on nonlocal and surface elasticity theories

This study employs the nonlocal and surface elasticity theories to explore the buckling characteristics of piezoelectric nanobeams incorporating flexoelectricity effects. Flexoelectricity represents the coupling between the strain gradients and electrical polarizations. Considering the flexoelectricity effects, the piezoelectric nanobeams can tolerate higher buckling loads compared with conventional ones, especially at lower thicknesses. Both nonlocal and surface effects are considered in the analysis of flexoelectrically actuated piezoelectric nanobeams in magnetic field for the first time. Hamilton’s principle is employed to derive the governing equations and the related boundary conditions which are solved applying an analytical-based solution. 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 parameter, surface elasticity, temperature rise, beam thickness and various boundary conditions on the buckling characteristics of magnetically affected flexoelectric nanobeam.

Nonlocal strain gradient-based vibration analysis of embedded curved porous piezoelectric nano-beams in thermal environment

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

Static and dynamic analytical coupled field analysis of piezoelectric flexoelectric nanobeams: A strain gradient theory approach

Piezoelectric and flexoelectric nanostructures attracted great interest from different research communities for their potential applications as sensors, actuators and energy harvesters. Nevertheless, modeling and analysis of such structures are still under investigation. In fact, strain gradient effects are expected to be nonnegligible at the nanoscale level where they induce substantial variations on the static and dynamic responses of the nanostructure, especially when coupled field behavior are present. Therefore, flexoelectricity which refers to the electromechanical coupling between electrical polarization and mechanical strain gradient, is expected to be dominant at the nanoscale. In this paper, the main focus is to analyze analytically the static and dynamic responses of nanobeams, having different boundary conditions and electrical loads, where gradient elasticity, piezoelectricity and flexoelectricity are taken into account. We develop a complete mathematical model using Hamilton’s principle. The derived governing electromechanical coupled equations and corresponding boundary conditions are solved using a Galerkin procedure based on an assumed mode approach. The principal electromechanical outputs are calculated analytically for actuation and energy harvesting configurations. Bidirectional polarization, electric field and elastic strain gradient effect are taken into account in the developed model. Validation and comparison with previously published results showed that considerable decrease of the performance could be observed because of the introduction of the elastic strain gradient effect. The performance degradation is also more pronounced if the aspect ratio is reduced.

Nonlocal and Surface Effects on Vibration Behavior of Axially Loaded Flexoelectric Nanobeams Subjected to In-Plane Magnetic Field

In this research, vibration analysis of axially preloaded nonlocal flexoelectric nanobeams considering surface and magnetic field effects is performed. The main propose of present paper is to fill the gap in previous analyzes of flexoelectric nanobeams in which nonlocal and surface effects are ignored. Flexoelectricity in a nanobeam results in larger vibration frequencies than a typical piezoelectric nanobeam. Via Galerkin’s technique, the governing equations of the flexoelectric nanobeam are numerically solved. The vibration frequency of the nanobeam is validated with previous one in the literature. The illustrative results demonstrate the influences of the flexoelectricity, nonlocal stress field, surface energy, magnetic field, axial load and beam thickness on vibrational characteristics of a nanosize preloaded flexoelectric beam.

Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory

Flexoelectric effect has been defined as the coupling between strain gradient and electric polarization, however, how to harvest remarkable polarization energy induced by flexoelectric effect is the key problem. The present work is to study the flexoelectric effect in functionally graded composite nanostructure with a volume fraction distribution function. And a more general modified strain gradient theory is used to reformulate the constitutive equations and a more scientific evaluation system is introduced to measure the electric polarization density field for static bending and free vibration behaviors of functionally graded piezoelectric nanobeams. Meanwhile, we put forward a new and simple volume fraction distribution function with two-parameters and the physical surface position for such nanobeams in which the material properties vary in the thickness direction is determined. Numerical results indicate that flexoelectric effect can observably influence the electromechanical response in functionally graded piezoelectric nanobeam at nanometer scale and the pertinent physical insights are also discussed. And the emerging functionally graded materials are significant and may help to resolve tantalizing application of flexoelectric effect on practical engineering.

On non-linear vibrations of flexoelectric nanobeams

In this article, non-linear vibration problem of flexoelectric nanobeams with surface and thermal effects is investigated. Flexolectricity in piezoelectric nanobeams can be interpreted as the strain gradient-electrical polarization coupling. There are two flexoelectric mechanisms which are direct and converse flexoelectricity. An electric voltage is applied to the top surface of the nanobeam which introduces a closed-circuit electric field condition. Employing He's variational method, the nonlinear governing equation of flexolectric nanobeam is solved and nonlinear frequencies are obtained. The flexielectricity effect is less important for large scale beams, therefore it has a great influence on vibration behavior of piezoelectric nanobeams. Also, the effects of several parameters including temperature change, elastic foundation, surface coefficients and nanobeam thickness on nonlinear frequency are investigated.

Investigating nonlinear vibration of closed circuit flexoelectric nanobeams with surface effects via Hamiltonian method

This paper is motivated by the lack a study on nonlinear vibration behavior of flexoelectric nanobeams. Nonlinear vibration analysis of a nanoscale piezoelectric beam resting on nonlinear elastic foundation under closed circuit condition is performed 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. Close form of nonlinear frequency is obtained implementing Hamiltonian method. Obtained results indicate that both the flexoelectricity and surface effects possess notable impact on the nonlinear frequencies of the system. Only flexoelectricity yields a considerable difference between the present model and previous investigations on conventional piezoelectric nanobeams. A parametric study has been performed to examine the effects of surface elasticity, flexoelectricity, applied electric voltage, foundation parameters, geometrical parameters and boundary conditions on nonlinear frequencies of piezoelectric nanobeams.

Size-dependent hygro–thermo–electro–mechanical vibration analysis of functionally graded piezoelectric nanobeams resting on Winkler–Pasternak foundation undergoing preload and magnetic field

In this study, nonlocal beam theory is utilized for vibration analysis of hygro–electro–thermo–mechanical of functionally graded material (FGM) nanobeam by consideration of magnetic field and preload. Moreover, the material properties are considered to vary corresponding to the thickness of nanobeam in the framework of power-law distribution. Differential equations are derived by means of Hamilton principle in the framework of Euler–Bernoulli beam theory. The derived governing differential equations are solved by differential transformation method (DTM) which demonstrates to have high precision and computational efficiency in the vibration analysis of nanobeams. Numerical results are presented for various boundary conditions. A detailed parametric study is conducted temperature to examine the effects of the nonlocal parameter, voltage and, elastic mediums, power-law index, aspect ratio, preload, magnetic field and moisture effect on vibration characteristics of functionally graded nanobeam. Numerical results are presented in this paper to serve as benchmarks for future analyses of nanotubes.

Nanoscale piezoelectric vibration energy harvester design

Development of new nanoscale devices has increased the demand for new types of small-scale energy resources such as ambient vibrations energy harvesters. Among the vibration energy harvesters, piezoelectric energy harvesters (PEHs) can be easily miniaturized and fabricated in micro and nano scales. This change in the dimensions of a PEH leads to a change in its governing equations of motion, and consequently, the predicted harvested energy comparing to a macroscale PEH. In this research, effects of small scale dimensions on the nonlinear vibration and harvested voltage of a nanoscale PEH is studied. The PEH is modeled as a cantilever piezoelectric bimorph nanobeam with a tip mass, using the Euler-Bernoulli beam theory in conjunction with Hamilton’s principle. A harmonic base excitation is applied as a model of the ambient vibrations. The nonlocal elasticity theory is used to consider the size effects in the developed model. The derived equations of motion are discretized using the assumed-modes method and solved using the method of multiple scales. Sensitivity analysis for the effect of different parameters of the system in addition to size effects is conducted. The results show the significance of nonlocal elasticity theory in the prediction of system dynamic nonlinear behavior. It is also observed that neglecting the size effects results in lower estimates of the PEH vibration amplitudes. The results pave the way for designing new nanoscale sensors in addition to PEHs.