Piezoelectric Nanobeams Research Articles

Modeling of size dependent buckling behavior of piezoelectric sandwich perforated nanobeams rested on elastic foundation with flexoelectricity

This article presents a size dependent mathematical model and an analytical solution methodology to accurately simulate and analyze the size dependent buckling behavior of piezoelectrically layered sandwich nanobeams with perforated core embedded in an elastic medium with flexoelectricity. The electric enthalpy energy function is expressed in terms of the electric and the flexoelectric effects. Regular squared cutouts perforation pattern is adopted for the perforated core. Closed forms for the equivalent geometrical parameters of perforated core are developed. The shear deformation effect is incroporated using the Timoshenko beam theory (TBT). To capture the nonlocality and the microstructure length scale effects, the nonlocal strain gradient elasticity theory is modified and adopted to include the electromechanical nonclassical effects. The Hamiltonian principle is utilized to derive the equilibrium equations. An analytical solution methodology is developed to derive closed forms for critical buckling loads. The developed solution procedure is implemented into a MATLAB software code. The accuracy of the developed procedure is verified by comparing obtained results with corresponding cases reported in the literature. Influences of different design variables on the electromechanical buckling behavior are explored through intensive parametric studies. Results obtained showed the significant effects of the flexoelectricity and the piezoelectric parameters on the size dependent buckling behavior of piezoelectric sandwich nanobeams with perforated core. Size dependent electromechanical as well as mechanical buckling behaviors could be controlled by adjusting these parameters. The developed procedure and the obtained numerical results are helpful in many technological and industrial applications as MEMS and NEMS.

Nonlinear analysis of unimorph piezoelectric nanobeam with a variable cross-section and flexoelectric effect

An analytical method is presented to investigate the nonlinear electromechanical coupling response of an unimorph piezoelectric nanobeam with a variable cross-section, taking into account the flexoelectric effect. Considering the geometric nonlinearity, strongly nonlinear governing equation with variable coefficients is derived using the energy variation method and solved by the differential quadrature method. The nonlinear effects of geometric nonlinearity, electromechanical loadings, flexoelectric effect and width ratio on the electromechanical coupling response are analyzed in detail. The results show that under different loadings, the flexoelectric parameter and width ratio have distinct effects on the electromechanical response of the piezoelectric variable cross-section nanobeam. When the mechanical loads are applied on the nanobeam, a larger flexoelelctric parameter results in a greater bending deflection and higher output of electric field and polarization. The width ratio of variable cross-section can also influence the bending deflection, but its impact on the output of electric field and polarization is limited. When the applied voltage loads are applied to the nanobeam, the bending direction can be controlled by the applied voltage, flexoelectric parameter or width ratio. Due to the strongly nonlinear governing equation, these results in this study differ significantly from those of piezoelectric nanobeam with uniform cross-section. This study can contribute to the understanding of electromechanical coupling response in piezoelectric variable cross-section nanobeam with flexoelectric effect. Additionally, it can be helpful in guiding the structure optimization of piezoelectric nanodevices.

Surface Elastic Effects on Electromechanical Responses of a Piezoelectric Semiconducting Nanobeam

Surface Elastic Effects on Electromechanical Responses of a Piezoelectric Semiconducting Nanobeam

Exact solution of thermal postbuckling of porous piezoelectric nanobeams with surface effect

In this paper, the thermal postbuckling of porous piezoelectric nanobeams is analyzed using the first-order shear deformation beam theory based on the surface elasticity theory and "core-shell" model, where the surface effect is introduced by employing the surface energy model. The governing equations for thermal postbuckling of porous piezoelectric nanobeams were derived by the principle of minimum potential energy. It is found that surface effect increases the effective elastic modulus and dimensionless critical load and reduces the thermal postbuckling conformations, thermal postbuckling paths and amount of induced charge. Meanwhile, the reasonable pore distribution pattern can effectively improve the mechanical properties.

Fractional modelling of piezoelectric composite nanobeams via novel numerical schemes

Fractional modelling of piezoelectric composite nanobeams via novel numerical schemes

Exact solution of post-buckling behavior of porous piezoelectric nanobeams with surface effects

Exact solution of post-buckling behavior of porous piezoelectric nanobeams with surface effects

Nonlinear Vibration and Stability Analysis of Rotating Functionally Graded Piezoelectric Nanobeams

Presented herein is an investigation for the nonlinear vibration and stability analysis of rotating functionally graded (FG) piezoelectric nanobeams based on the nonlocal strain gradient theory. The present model can be regarded as a simplified version for the rotating nanowire of biomechanical nanogenerators. The Hamilton principle is used to derive nonlinear equations of motion and their related boundary conditions, which are then discretized to form a set of algebraic equations. Accordingly, the nonlinear vibration frequencies and buckling loads of the nanobeams can be determined by an iterative method. A parametric study of rotational velocity, nonlocal parameter, material length parameter, power-law index, and electrostatic voltage on the dynamic stability behavior of such nanobeams is also presented. In the cantilever case, increasing the nonlocal parameter and material length parameter can result in a stiffness-hardening effect that is unaffected by rotational velocity and the material length parameter to nonlocal parameter ratio. Yet, this has not been reported previously. More importantly, incorporating the effect of geometric nonlinearity on the dynamic responses and stability results of the nanobeams is indispensable. In particular, new observations for the coupling effect of vibration amplitude and power-law index on the electric potential effect are useful for the design of rotating microelectromechanical devices.

Tunability of “spring-mass” resonator on band gaps with surface effects of a piezoelectric phononic crystal double-layer nanobeam

A model of piezoelectric phononic crystal (PC) double-layer nanobeam periodically attached by “spring-mass” resonators is proposed. As a comparison, two other simplified models are also proposed. In order to calculate the band structure, the plane wave expansion (PWE) method is improved by introducing the Euler beam and surface piezoelectricity theories. All the results and analysis demonstrate that the tunable advantage is displayed in double-layer nanobeam on account of the antisymmetric and symmetric vibrational modes controlled by adding additional springs between two different nanobeams. Moreover, the influencing rules of parameters related to additional spring and “spring-mas” resonator are also revealed. The bandgap properties can be applied to nano electro-mechanical system (NEMS) to realize active control of vibration.

Influence of surface effect on post-buckling behavior of piezoelectric nanobeams

Piezoelectric nanobeams with excellent mechanical, thermal and electrical properties are important components in micro-nano electromechanical systems, which are widely used as sensors, brakes and resonators. Based on the Euler–Bernoulli beam model, the influence of surface effect on the post-buckling behaviour of piezoelectric nanobeams is analysed. According to the surface elasticity theory and the ‘core–shell’ model, the surface energy model is used to introduce the influence of surface effect. The governing equations and boundary conditions of the post-buckling of piezoelectric nanobeams under the influence of surface effect are derived by the principle of minimum potential energy. The analytical solution of post-buckling is obtained by the eigenvalue method. The influence of surface effect on the post-buckling configuration, post-buckling path, amount of induced charge and critical load of piezoelectric nanobeams with different external constraints and cross-sectional dimensions are discussed. The results show that surface effect has a significant influence on the post-buckling of piezoelectric nanobeams. Considering surface effect, the effective elastic modulus and critical load of piezoelectric nanobeams are increased, and the post-buckling configuration, post-buckling path and amount of induced charge are reduced. These findings contribute to the study of micro-nano electromechanical systems based on nanobeam structures and provide a theoretical basis for the design and manufacture of nanodevices.

The electro-mechanical coupling responses of functionally graded piezoelectric nanobeams with flexoelectric effect

The aim of this paper is to study the electro-mechanical coupling responses of functionally graded (FG) piezoelectric cantilever nanobeam under the concentrated load, and the material properties follow an exponential distribution in the thickness direction. The constitutive equations are derived from the electric Gibbs free energy density function. The governing equations and boundary conditions of the Euler–Bernoulli FG beam can be obtained by variational principle, and then, the analytical expressions for its deflection, polarization intensity, and induced potential are obtained. The results show that the flexoelectric effect has a significant impact on the deflection, polarization intensity, and induced potential of the FG cantilever nanobeam.

Nonlocal state-space strain gradient wave propagation of magneto thermo piezoelectric functionally graded nanobeam

Abstract In this work, the state -space nonlocal strain gradient theory is used for the vibration analysis of magneto thermo piezoelectric functionally graded material (FGM) nanobeam. An analysis of FGM constituent properties is stated by using the power law relations. The refined higher order beam theory and Hamilton’s principle have been used to obtain the motion equations. Besides, the governing equations of the magneto thermo piezoelectric nanobeam are extracted by developed nonlocal state-space theory. And to solve the wave propagation problems, the analytical wave dispersion method is used. The effect of magnetic potential, temperature gradient, and electric voltage in variant parameters are presented in graph.

Investigation of Size-Dependent Vibration Behavior of Piezoelectric Composite Nanobeams Embedded in an Elastic Foundation Considering Flexoelectricity Effects

This article investigates the size dependent on piezoelectrically layered perforated nanobeams embedded in an elastic foundation considering the material Poisson’s ratio and the flexoelectricity effects. The composite beam is composed of a regularly squared cut-out elastic core with two piezoelectric face sheet layers. An analytical geometrical model is adopted to obtain the equivalent geometrical variables of the perforated core. To capture the Poisson’s ratio effect, the three-dimensional continuum mechanics adopted to express the kinematics are kinetics relations in the framework of the Euler–Bernoulli beam theory (EBBT). The nonlocal strain gradient theory is utilized to incorporate the size-dependent electromechanical effects. The Hamilton principle is applied to derive the nonclassical electromechanical dynamic equation of motion with flexoelectricity impact. A closed form solution for resonant frequencies is obtained. Numerical results explored the impacts of geometrical and material characteristics on the nonclassical electromechanical behavior of nanobeams. Obtained results revealed the significant effects of the mechanical, electrical, and elastic foundation parameters on the dynamic behavior of piezoelectric composite nanobeams. The developed procedure and the obtained results are helpful for many industrial purposes and engineering applications, such as micro/nano-electromechanical systems (MEMS) and NEMS.

On Bending of Piezoelectrically Layered Perforated Nanobeams Embedded in an Elastic Foundation with Flexoelectricity

Analysis of the electromechanical-size-dependent bending of piezoelectric composite structural components with flexoelectricity has been considered by many researchers because of the developments of nanotechnology and the applicability of piezoelectric composite nanobeam structures in Micro/Nano-Electro-Mechanical Systems (MEMS/NEMS). Therefore, the work investigates the size-dependent electromechanical bending of piezoelectrically layered perforated nanobeams resting on elastic foundations including the flexoelectric effect. Within the framework of the modified nonlocal strain gradient elasticity theory, both the microstructure and nonlocality effects are captured. The governing equilibrium equations including piezoelectric and flexoelectric effects are derived using Hamilton’s principle. Closed forms for the non-classical electromechanical bending profiles are derived. The accuracy of the proposed methodology is verified by comparing the obtained results with the available corresponding results in the literature within a 0.3% maximum deviation. Parametric studies are conducted to explore effects of perforation parameters, elastic foundation parameters, geometric dimensions, nonclassical parameters, flexoelectric parameters, as well as the piezoelectric parameters on the bending behavior of piezoelectrically layered perforated nanobeams. The obtained results demonstrate that incorporation of the nondimensional elastic foundation parameters, Kp = 2 and Kw = 20, results in a reduction in the relative percentage reduction in the maximum nondimensional mechanical transverse deflection due to increasing the perforation filling ratio from 0.2 to 1 from 199.86% to 91.83% for a point load and 89.39% for a uniformly distributed load. On the other hand, with Kp = 5 and Kw = 50, the relative percentage difference of the electromechanical bending deflection due to increasing the piezoelectric coefficient, e311, reaches about 8.7% for a point load and 8.5% for a uniformly distributed load at a beam aspect ratio of 50. Thus, the electromechanical as well as mechanical behaviors could be improved by controlling these parameters. The proposed methodology and the obtained results are supportive in many industrial and engineering applications, i.e., MEMS/NEMS.

Vibration of a piezoelectric nanobeam with flexoelectric effects by Adomian decomposition method

Vibration of a piezoelectric nanobeam with flexoelectric effects by Adomian decomposition method

Bandgap characteristics of a piezoelectric phononic crystal Timoshenko nanobeam based on the modified couple stress and surface energy theories

This paper establishes an analytical model for the size-dependent flexural wave band structures of a piezoelectric phononic crystal (PC) Timoshenko nanobeam, considering the shear deformation and rotational inertia effects. The governing equation is derived based on the surface piezoelectricity energy and modified couple stress theory in the bulk and surface layer of the piezoelectric PC nanobeam, respectively. As a main objective of this article, the surface effects including the surface elasticity, residual surface stress and surface piezoelectricity on the frequency bandgaps in the wave propagation band structures are investigated for a piezoelectric PC Timoshenko nanobeam. Also, the effects of some geometrical parameters on the band structures are studied in details. An analytical approach is proposed for the problem. The obtained results for a piezoelectric PC Timoshenko nanobeam are compared with those of previous researches for a piezoelectric PC Euler-Bernoulli nanobeam. It is concluded that the geometrical parameters and residual surface elasticity affect the location and width of bandgaps in the frequency band structures. Schematic configuration of a piezoelectric PC Timoshenko nanobeam with a surface layer and rectangular cross section.

Vibration analysis of the rotating multilayer piezoelectric Timoshenko nanobeam

The vibration behavior of multilayers piezoelectric nanobeam is investigated in this study. The governing equations of rotating piezoelectric beams are obtained by utilizing the Timoshenko beam theory. The nonlocal continuum theory along with surface elasticity theory are handled to derive the differential governing equation of motion. The differential governing equation of motions is solved by employing the DQM. The results show a great correlation between the numerical results and the presented results in the literature. The results are obtained for four different boundary conditions such as cantilever nanobeam (clamped-free or CF), simply-simply (SS), clamped-simply (CS), and clamped-clamped (CC). The reaction of vibration frequencies to the change of different parameters such as the external voltage, the thermal stress, the angular velocity, etc. is investigated. The results show that the boundary condition type, thermal stress, and external voltage have a vital consequence on the vibration frequency. The results of this study can be appropriated to design and fabricated the nanodevices such as nanomotors, nano turbines, biosensors, nano-electromechanical systems, and micro-electromechanical systems (NEMS/MEMS) devices.

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

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

Free transverse vibrations analysis of size-dependent cracked piezoelectric nano-beam based on the strain gradient theory under mechanic-electro forces

This paper investigates the free lateral vibration of a cracked nano-beam to piezoelectric effects on Euler-Bernoulli beam theory and nonlocal strain gradient theory (NSGT). So far, the propagation and growth of cracks in nano-beams have not been done by modeling the crack area using torsion springs using NSGT. This article investigates the free vibrations of cracked nano-beams using piezoelectric effects. The governing equations, related boundary conditions, and piezoelectric effects were derived using the Hamilton principle. Nano-beam separation is modeled by splitting nanoparticles into two parts connected to a rotating spring. The model calls the extra strain energy due to cracking and increases the discontinuity in the deflection slope. This study investigates the effects of crack propagation, crack intensity, piezoelectric effects, longitudinal scale parameter, and nonlocal parameter. A comparison of previous studies has been published in which a good agreement is observed. The results show that the mentioned parameters play an important role in dynamic behavior.

Dynamic Analysis of a Piezoelectrically Layered Perforated Nonlocal Strain Gradient Nanobeam with Flexoelectricity

This study presents a mathematical size-dependent model capable of investigating the dynamic behavior of a sandwich perforated nanobeam incorporating the flexoelectricity effect. The nonlocal strain gradient elasticity theory is developed for both continuum mechanics and flexoelectricity. Closed forms of the equivalent perforated geometrical variables are developed. The Hamiltonian principle is exploited to derive the governing equation of motion of the sandwich beam including the flexoelectric effect. Closed forms for the eigen values are derived for different boundary conditions. The accuracy of the developed model is verified by comparing the obtained results with the available published results. Parametric studies are conducted to explore the effects of the perforation parameters, geometric dimensions, nonclassical parameters, flexoelectric parameters, as well as the piezoelectric parameters on the vibration behavior of a piezoelectric perforated sandwich nanobeam. The obtained results demonstrate that both the flexoelectric and piezoelectric parameters increased the vibration frequency of the nanobeam. The nonlocal parameter reduced the natural vibration frequency due to a decrease in the stiffness of the structures. However, the strain gradient parameter increased the stiffness of the structures and hence increased the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the ration of the value of the nonlocal parameter to the strain gradient parameter. This model can be employed in the analysis and design of NEMS, nanosensors, and nanoactuators.

On well-posed integral nonlocal gradient piezoelectric models for static bending of functionally graded piezoelectric nanobeam

In this paper, we propose general strain- and stress-driven integral nonlocal gradient piezoelectric models (INGPM), which are capable of distinguishing the softening and toughening size-effects due to elasticity and piezoelectricity. Based on differential calculation, INGPM can be transformed equivalently to differential forms with four constitutive constraints. Due to absence the nonlocal and gradient data, the simplified INGPMs with identical elastic and piezoelectric size-dependent parameters are applied to formulate the static bending of functionally graded piezoelectric Euler-Bernoulli nanobeam. The governing differential equations and boundary conditions as well as constitutive constraints are deduced and expressed in nominal forms. It is found that both strain- and stress-driven INGPMs would lead to well-posed mathematical formulation because the total differential order of the differential equations equal to the number of boundary conditions and constitutive constraints. Neglecting the constitutive constraints, the strain-driven INGPM turns to be differential nonlocal gradient piezoelectric model (DNGPM), which is widely applied to study the size-effect of nanostructures. However, it shows that DNGPM would result in ill-posed mathematical formulation, because the total number (12) of differential orders for governing equations is larger than the number (6) of boundary conditions. The general differential quadrature method (GDQM) is utilized to discretize the differential governing equations, boundary conditions and constitutive constraints. Numerical results show that bending deflections increase consistently with the increase of nonlocal parameter and the decrease of gradient parameter for strain-driven INGPM and the decrease of nonlocal parameter and the increase of gradient parameter for stress-driven INGPM, respectively.