Flexoelectricity, representing the coupling between electrical polarizations and strain gradients, should be taken into account in the analysis of electromechanical responses of nanostructures where large strain gradients are expected. In this paper, we will explore the influence of flexoelectricity on the electromechanical coupling behavior of a simply supported piezoelectric nanoplate by using the Kirchhoff plate theory. The governing equations and corresponding boundary conditions are deduced from Hamilton’s principle, and the analytical solutions are obtained for the deflection and natural frequency. The results indicate that the deflections predicted by the present model are smaller than those calculated by the classical one which only considers piezoelectricity, while the frequencies exhibit the opposite trend. In addition, the flexoelectric effect is more prominent for thinner plates; the differences of the deflections or frequencies between the two models are gradually diminishing with an increase in the plate thickness. The current work may contribute to the understanding of the higher-order electromechanical coupling mechanism. Moreover, the modified plate model can be utilized to accurately design novel piezoelectric nanoplate-based sensors in nanoelectromechanical systems.

Electromechanical responses of piezoelectric nanoplates with flexoelectricity

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

This paper deals with the size-dependent geometrically nonlinear free vibration of magneto-electro-thermo elastic (METE) nanoplates using the nonlocal elasticity theory. The mathematical formulation is developed based on the first-order shear deformation plate theory, von Karman-type of kinematic nonlinearity and nonlocal elasticity theory. The influences of geometric nonlinearity, rotary inertia, transverse shear deformation, magneto-electro-thermal loading and nonlocal parameter are considered. First, the generalized differential quadrature (GDQ) method is utilized to reduce the nonlinear partial differential equations to a system of time-dependent nonlinear ordinary differential equations. Afterwards, the numerical Galerkin method, periodic time differential operators and pseudo-arc length continuation algorithm are employed to compute the nonlinear frequency versus the amplitude for the METE nanoplates. The presented methodology enables one to describe the large-amplitude vibration characteristics of METE nanoplates with various sets of boundary conditions. A detailed parametric study is carried out to analyze the important parameters such as the nondimensional nonlocal parameter, external electric potential, external magnetic potential, temperature change, length-to-thickness ratio, aspect ratio and various edge conditions on the nonlinear free vibration characteristics of METE nanoplates. The results demonstrate that considering the size effect on the vibration response of METE nanoplate results in decreasing the natural frequency, a remarkable increasing effect on the hardening behavior and subsequently increasing the nonlinear-to-linear frequency ratio.

Size-Dependent Nonlinear Vibrations of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

This paper studies stress distribution in a single-walled carbon nanotube (SWCNT) under internal pressure with various chirality. Strain gradient theory is used to capture the size-dependent behavior of the SWCNT. Minimum total potential energy principle is successfully applied to derive the governing differential equation and its associated boundary conditions. Due to complexity of the governing differential equation and boundary conditions, numerical scheme is used to solve the problem. Comparing the results based on strain gradient theory and that of classical elasticity shows a major difference between these two methods. However, a close examination of the results indicates that both theories predict the same trend for variations in the radial displacement along the SWCNT radius. Numerical results also indicate that the proposed model can lead into the classical elasticity model, provided the material length scale parameters are taken to be zero. Additionally, for plane strain condition, the radial di...

Size-Dependent Stress Analysis of Single-Wall Carbon Nanotube Based on Strain Gradient Theory

In flexoelectricity, a strain gradient can induce polarization and a polarization gradient can induce mechanical stress. In this paper, in order to identify the contributions of each strain gradient component, the flexoelectric theory is reformulated by splitting the strain gradient tensor into mutually independent parts. Two sets of orthogonal higher-order deformation metrics are inherited and perfected to reformulate the internal energy density for isotropic materials. The deviatoric stretch gradient and the symmetric part of the rotation gradient are proved to disappear in the coupling of strain gradient to polarization and, moreover, the independent higher-order constants associated with the coupling of strain gradient to strain gradient reduce from five to three. The constitutive relations are then reformulated in terms of the new deformation and electric field metrics, and the governing equations and boundary conditions are derived according to the variational principle of electric enthalpy. On the basis of the present simplified flexoelectric theory, a flexoelectric Bernoulli–Euler beam theory is specified. Solutions for a cantilever subjected to a force at the free end and a voltage cross the thickness are constructed and the size-dependent direct and inverse flexoelectric effects are captured.

A reformulated flexoelectric theory for isotropic dielectrics

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.

Liang Xu

Papers

Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity