Size-dependent Piezoelectricity Research Articles

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.

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.

Electrostatic potential in a bent piezoelectric nanowire with consideration of size-dependent piezoelectricity and semiconducting characterization

Determining the electric potential in a bent piezoelectric nanowire (NW) is a fundamental issue of nanogenerators and nanopiezotronics. The combined influence of the flexoelectric effect, the semiconducting performance and the angle of atomic force microscope (AFM) tip has never been studied previously and will be investigated in this paper. The exact solution for the electric potential of a bent piezoelectric semiconductor NW is derived. The electric potential of the present model with consideration of flexoelectric effect varies along the length of the NW and is different from that of the classical piezoelectric model. Flexoelectric effect enhances but the semiconducting performance reduces the electric potential of the NW. In addition, it is found that if the angle of the AFM tip reaches 30°, the error of the electric potential obtained from the model ignored the effect of the angle of the AFM tip is almost 16%, which is unacceptable.

Size-dependent piezoelectricity of molybdenum disulfide (MoS2) films obtained by atomic layer deposition (ALD)

As a member of transition metal dichalcogenides, MoS2 is an ideal low-dimensional piezoelectric material, which makes it attract wide attention for potential usage in next generation piezoelectric devices. In this study, the size-dependent piezoelectricity of MoS2 films with different grain sizes obtained at different temperatures by atomic layer deposition (ALD) was determined, which indicates that the grain size is critical to the piezoelectric constant. When the grain size is less than 120 nm, the piezoelectric constant increases with the increase in the grain size. Moreover, the piezoelectric constant first increases and then decreases with the increase in the film thickness. Therefore, piezoelectric constants of these MoS2 films can be modulated by changing the growth temperature and applying different ALD cycles.

A Nonlinear Electro-Mechanical Analysis of Nanobeams Based on the Size-Dependent Piezoelectricity Theory

AbstractNonlinear formulation of isotropic piezoelectric Euler-Bernoulli nano-beam is developed based on consistent size-dependent piezoelectricity theory. By considering geometrically nonlinear and axial displacement of the centroid of beam sections, basic nonlinear equations of piezoelectric nanobeam are derived using Hamilton's principle and variational method. Afterwards, in the special case for the formulation derived, hinged-hinged piezoelectric nanobeam is studied, and static deflection as well as free vibrations of the nanobeam under mechanical loads is determined. In this case, results of the linear formulation of the size-dependent theory are compared to those of the linear and nonlinear classical continuum theory.

Size-dependent analysis of piezoelectric nanobeams including electro-mechanical coupling

In this paper, nonlinear formulation of the size-dependent piezoelectric Timoshenko nano-beam is developed based on the consistent size-dependent piezoelectricity theory. Nonlinear basic equations as well as linear and nonlinear boundary conditions of the piezoelectric nanobeam are derived using Hamilton’s principle and the variational method. To derive these equations, the von Karman strain is employed to model the nonlinear geometric model for the nanobeam behavior. To evaluate the formulation derived, static deformation and free vibration of the hinged-hinged piezoelectric beam is investigated in the special case. The results of the formulation derived are investigated under different parameters, and linear and nonlinear effects of the new size-dependent formulation with the classical theory are compared.

Boundary element formulation for plane problems in size-dependent piezoelectricity

A new boundary element formulation is developed to analyze two-dimensional size-dependent piezoelectric response in isotropic dielectric materials. The model is based on the recently developed consistent couple stress theory, in which the couple-stress tensor is skew-symmetric. For isotropic materials, there is no classical piezoelectricity, and the size-dependent piezoelectricity or flexoelectricity effect is solely the result of coupling of polarization to the skew-symmetric mean curvature tensor. As a result, the size-dependent effect is specified by one characteristic length scale parameter l, and the electromechanical effect is specified by one flexoelectric coefficient f. Interestingly, in this size-dependent multi-physics model, the governing equations are decoupled. However, the problem is coupled, because of the existence of a flexoelectric effect in the boundary couple-traction and normal electric displacement. We discuss the boundary integral formulation and numerical implementation of this size-dependent piezoelectric boundary element method, which provides a boundary-only formulation involving displacements, rotations, force-tractions, couple-tractions, electric potential, and normal electric displacement as primary variables. Afterwards, we apply the resulting BEM formulation to several computational problems to confirm the validity of the numerical implementation and to explore the physics of the flexoelectric coupling. Copyright © 2016 John Wiley & Sons, Ltd.

Size-dependent piezoelectricity: A 2D finite element formulation for electric field-mean curvature coupling in dielectrics

The classical theory of piezoelectricity defines linear size-independent electromechanical response in non-centrosymmetric dielectrics that involves coupling between the electric field and the mechanical strains. However, with the continuing push to develop novel micro- and nano-scale materials, structures and devices, there is a need to refine and explore size-dependent electro-mechanical coupling phenomena, which have been observed in experiments on centrosymmetric dielectrics. Here a finite element variational formulation is developed based upon a recent consistent size-dependent theory that incorporates the interactions between the electric field and the mechanical mean curvatures in dielectrics, including those with centrosymmetric structure. The underlying formulation is theoretically consistent in several important aspects. In particular, the electric field equations are consistent with Maxwell's equations, while the mechanical field equations are based upon the recent consistent couple stress theory, involving skew-symmetric mean curvature and couple stress tensors . This, in turn, permits the development of a fully-consistent finite element method for the solution of size-dependent piezoelectric boundary value problems. In this paper, an overview of size-dependent piezoelectricity is first provided, followed by the development of the variational formulation and finite element representation specialized for the planar response of centrosymmetric cubic and isotropic materials. The new formulation is then applied to several illustrative examples to bring out important characteristics predicted by this consistent size-dependent piezoelectric theory. • We develop a finite element method for consistent size-dependent piezoelectricity. • Our formulation is based on skew-symmetric mean curvature and couple stress tensors. • Unlike existing formulations, our theory is consistent with Maxwell's equations. • We employ Lagrange multipliers to impose displacement-rotation compatibility. • We solve several illustrative problems to reveal key characteristics of the theory.

SIZE-DEPENDENT PIEZOELECTRICITY AND ELASTICITY DUE TO THE ELECTRIC FIELD-STRAIN GRADIENT COUPLING AND STRAIN GRADIENT ELASTICITY

A size-dependent nonclassical Bernoulli–Euler beam model based on the strain gradient elasticity is proposed for piezoelectric nanowires. The governing equations and the corresponding boundary conditions are naturally derived from the variational principle. Different from the classical piezoelectric beam theory, the electric field–strain gradient coupling and the strain gradient elasticity are both taken into account. Static bending problem of a cantilever piezoelectric nanobeam is solved to illustrate the effect of strain gradient. The present model contains material length scale parameters and can capture the size dependent piezoelectricity and elasticity for nanoscale piezoelectric structures. The numerical results reveal that the deflections predicted by the present model are smaller than that by the classical beam theory and the effective electromechanical coupling coefficient is dramatic enhanced by the electric field–strain gradient coupling effect. However, the differences in both the deflections and effective EMC coefficient between the two models are very significant when the beam thickness is very small; they are diminishing with the increase of the beam thickness. This model is helpful for understanding the electromechanically coupling mechanism and in designing piezoelectric nanowires based devices.

Size-dependent piezoelectricity

In this paper, a consistent theory is developed for size-dependent piezoelectricity in dielectric solids. This theory shows that electric polarization can be generated as the result of coupling to the mean curvature tensor, unlike previous flexoelectric theories that postulate such couplings with other forms of curvature and more general strain gradient terms ignoring the possible couple-stresses. The present formulation represents an extension of recent work that establishes a consistent size-dependent theory for solid mechanics. Here by including scale-dependent measures in the energy equation, the general expressions for force- and couple-stresses, as well as electric displacement, are obtained. Next, the constitutive relations, the uniqueness theorem and the reciprocal theorem for the corresponding linear small deformation size-dependent piezoelectricity are developed. As with existing flexoelectric formulations, one finds that the piezoelectric effect can also exist in isotropic materials. However, in the present theory there is only one flexoelectric constant for isotropic material and the coupling is strictly through the skew-symmetric mean curvature tensor. In the last portion of the paper, this isotropic case is considered in detail by developing the corresponding boundary value problem for two dimensional analyses and obtaining a closed form solution for an isotropic dielectric cylinder.

Flexoelectric charge separation and size dependent piezoelectricity in dielectric solids

Flexoelectric charge separation and the associated size dependent piezoelectricity are investigated in centrosymmetric dielectric solids. Direct piezoelectricity can exist as external mechanical stress is applied to non-piezoelectric dielectrics with shapes such as truncated pyramids, due to elastic strain gradient induced flexoelectric polarization. Effective piezoelectric coefficient is analyzed in truncated pyramids, which is strongly enhanced by size reduction and depends on flexoelectricity, elastic compliance, and aspect ratio of the non-piezoelectric dielectric solids.

Erratum: Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect [Phys. Rev. B77, 125424 (2008)

This note corrects some errors in a previously published paper: Size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect[PHYSICAL REVIEW B 77, 125424 (2008)]. The comlete published paper is also included in this note.

Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect

Crystalline piezoelectric dielectrics electrically polarize upon application of uniform mechanical strain. Inhomogeneous strain, however, locally breaks inversion symmetry and can potentially polarize even nonpiezoelectric (centrosymmetric) dielectrics. Flexoelectricity\char22{}the coupling of strain gradient to polarization\char22{}is expected to show a strong size dependency due to the scaling of strain gradients with structural feature size. In this study, using a combination of atomistic and theoretical approaches, we investigate the ``effective'' size-dependent piezoelectric and elastic behavior of inhomogeneously strained nonpiezoelectric and piezoelectric nanostructures. In particular, to obtain analytical results and tease out physical insights, we analyze a paradigmatic nanoscale cantilever beam. We find that in materials that are intrinsically piezoelectric, the flexoelectricity and piezoelectricity effects do not add linearly and exhibit a nonlinear interaction. The latter leads to a strong size-dependent enhancement of the apparent piezoelectric coefficient resulting in, for example, a ``giant'' 500% enhancement over bulk properties in $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ for a beam thickness of $5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. Correspondingly, for nonpiezoelectric materials also, the enhancement is nontrivial (e.g., 80% for $5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ size in paraelectric $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ phase). Flexoelectricity also modifies the apparent elastic modulus of nanostructures, exhibiting an asymptotic scaling of $1∕{h}^{2}$, where $h$ is the characteristic feature size. Our major predictions are verified by quantum mechanically derived force-field-based molecular dynamics for two phases (cubic and tetragonal) of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$.

Size-dependent piezoelectricity in zinc oxide nanofilms from first-principles calculations

The size dependence of the piezoelectricity in ZnO nanofilms is investigated using ab initio density-functional theory calculations. The effective piezoelectric constant of ZnO nanofilms increases monotonically with increasing film thickness in the nanoscale simulated in the present work, and surprisingly, exceeds that of the bulk ZnO when the thickness is greater than 2.4nm. The enhancement over the bulk value reaches 11% when the film thickness is 2.9nm.