Electromechanical fields in a hollow piezoelectric cylinder under non-uniform load: flexoelectric effect

The relations of a local gradient non-ferromagnetic electroelastic continuum are used to solve the problem of an axisymmetrical loaded hollow cylinder. Analytical solutions are obtained for tetragonal piezoelectric materials of point group 4 mm for two cases of external loads applied to the body surfaces. Namely, the hollow pressurized cylinder and a cylinder subjected to an electrical voltage V across its thickness are considered. The derived solutions demonstrate that the non-uniform electric load causes a mechanical deformation of piezoelectric body, and vice versa, the inhomogeneous radial pressure of the cylinder induces its polarization. Such a result is obtained due to coupling between the electromechanical fields and a local mass displacement being considered. In the local gradient theory, the local mass displacement is associated with the changes to a material’s microstructure. The classical theory does not consider the effect of material microstructure on the behavior of solid bodies and is incapable of explaining the mentioned phenomena. It is also shown that the local gradient theory describes the size-dependent properties of piezoelectric nanocylinders. Analytical solutions to the formulated boundary-value problems can be used in conjunction with experimental data to estimate some higher-order material constants of the local gradient piezoelectricity. The obtained results may be useful for a wide range of appliances that utilize small-scale piezoelectric elements as constituting blocks.

Local gradient Bernoulli–Euler beam model for dielectrics: effect of local mass displacement on coupled fields

The size-dependent behaviour of a Bernoulli–Euler nanobeam based on the local gradient theory of dielectrics is investigated. By using the variational principle, the linear stationary governing equations of the local gradient beam model and corresponding boundary conditions are derived. In this set of equations the coupling between the strain, the electric field and the local mass displacement is taken into account. Within the presented theory, the process of local mass displacement is associated with the non-diffusive and non-convective mass flux related to the changes in the material microstructure. The solution to the static problem of an elastic cantilever piezoelectric beam subjected to end-point loading is used to investigate the effect of the local mass displacement on the coupled electromechanical fields. The obtained solution is compared to the corresponding ones provided by the classical theory and strain gradient theory. It is shown that the beam deflection predicted by the local gradient theory is smaller than that by the classical Bernoulli–Euler beam theory when the beam thickness is comparable to the material length-scale parameter. The obtained results also indicate that the piezoelectricity has a significant influence on the electromechanical response in a dielectric nanobeam. The presented mathematical model of the dielectric beam may be useful for the study of electromechanical coupling in small-scale piezoelectric structures.