A flexoelectric theory with rotation gradient effects for elastic dielectrics

In this paper, a general flexoelectric theory in the framework of couple stress theory is proposed for isotropic dielectrics, in which the rotation gradient and the polarization gradient are involved to represent the nonlocal mechanical and electrical effects, respectively. The present flexoelectric theory shows only the anti-symmetric part of rotation gradient can induce polarization, while the symmetric part of rotation gradient cannot induce polarization in isotropic dielectrics. The electrostatic stress is obtained naturally in the governing equations and boundary conditions in terms of the variational principle, which is composed of two parts: the Maxwell stress corresponding to the polarization and the remainder relating to the polarization gradient. The current theory is able to account for the effects of size, direct and inverse flexoelectricities, and electrostatic force. To illustrate this theory, a simple application of Bernoulli-Euler cantilever beam is discussed. The numerical results demonstrate neither the higher-order constant l ₁ nor the higher-order constant l ₂ associated with the symmetric and anti-symmetric parts of rotation gradient, respectively, can be ignored in the flexoelectric theory. In addition, the induced deflection increases as the increase of the flexoelectric coefficient. The polarization is no longer constant and the potential is no longer linear along the thickness direction of beam because of the influence of polarization gradient.