A nonlocal variational principle for the converse flexoelectric effect based on simplified strain gradient elasticity

In this paper, we investigate how displacement variations affect electrical internal state variables, including polarization, polarization gradients, and the electric field. Our analysis is based on a hypothesis initially proposed by Stratton and later refined by Landau and Lifshitz. We focus on the converse flexoelectric effect in isotropic elastic dielectric materials within the framework of the simplified strain gradient theory of elasticity. This approach, combined with the variational principle applied to the strain energy functional and the virtual work of external forces, introduces nonlinearity into the equation of motion for the mathematical model, while the electrostatic equations governing polarization and its gradient remain linear. The direct solution method cannot obtain an analytical solution in the presence of nonlinearity; therefore, we turn to the exponential reductive perturbation method (ERPT) to obtain hierarchical solutions for the problem. Three cases will be analyzed, comparing the lattice parameter of the internal structure of STO with the length scale parameter used in the mathematical model. The results will be plotted and discussed in detail.