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.