Variational principles and governing equations in nano-dielectrics with the flexoelectric effect

Abstract

The flexoelectric effect is very strong and coupled with large strain gradients for nanoscale dielectrics. At the nanoscale, the electrostatic force cannot be ignored. In this paper, we have established the electric enthalpy variational principle for nanosized dielectrics with the strain gradient and the polarization gradient effect, as well as the effect of the electrostatic force. The complete governing equations, which include the effect of the electrostatic force, are derived from this variational principle, and based on the principle the generalized electrostatic stress is obtained, the generalized electrostatic stress contains the Maxwell stress corresponding to the polarization and strain, and stress related to the polarization gradient and strain gradient. This work provides the basis for the analysis and computations for the electromechanical problems in nanosized dielectric materials. flexoelectric effect, variational principle, governing equations, dielectrics, electrostatic stress, nanoscale PACS: 46.05.+b, 46.25.Hf, 62.25.+g, 68.35.Md

Flexoelectric effect on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate

“…When the PNP is subjected to an electric potential U across the plate thickness, the electric field is assumed to exist only along the thickness direction as adopted by Zhao et al, 33 which is expressed in terms of the electric potential U and the higher order local electrical force E ij , 27 as…”

Flexoelectric Effect on Vibration of Piezoelectric Microbeams Based on a Modified Couple Stress Theory

“…In Figure 3(a), where only piezoelectric effect is considered (i.e., without flexoelectricity), = 1.461 is obtained at = 0 showing that the size effect increases by 46.1% for all PMBs of different length and length scale parameter . Equations (13) and (14) show the contribution of the size effect ( ) on the effective bending stiffness ( ) eff of the PMBs. At = 20 m, increases to 1.465 when = −2 V is applied whereas it deceases to 1.457 at = 2 V. This result shows that the external voltage leads to the increase (or decrease) of the frequency of PMBs in the presence of a negative (or positive) voltage.…”

“…Flexoelectricity [4][5][6][7] is, however, the coupling between the mechanical strain gradient and the electric polarization, and it is a universal electromechanical mechanism in all insulators including piezoelectric materials [8][9][10][11]. Shen and Hu [12,13] introduced an electric Gibbs function to consider both of the piezoelectricity and flexoelectricity and derived the governing equations for dielectric materials. Flexoelectricity has also been found to be useful when it is used for sensing purpose [14,15].…”

A theory of flexoelectricity with surface effect for elastic dielectrics

“…Then, after a lengthy derivation (the details of which can be found in Hu and Shen (2010) and in the Appendix of this paper), the variation in bulk electric enthalpy can be expressed as…”