Nonlinear Modeling of Ferroelectric Plates as Electro-Elastic Material Surfaces

Abstract

We discuss nonlinear modeling of ferroelectric plates as electro-elastic material surfaces assuming the plate as a two-dimensional continuum with five mechanical degrees of freedom for each material point. Constitutive coupling by means of electrostriction, ferroelectric hysteresis and piezoelectricity are accounted for. For that sake, the augmented free energy is additively decomposed into an elastic part, a dielectric part and an augmentation energy. The elastic part involves an additive decomposition of the plate strain measures into elastic and electrical parts. Here, the electrical parts account for electrostriction and piezoelectricity, where electrostriction is assumed to depend quadratically on the polarization. For the dielectric part, we introduce the internal energy as a function of the polarization and an internal polarization, which allows us to accurately capture nonlinearities, such as saturation. Then, we compute the free energy applying a Legendre transformation, such that the voltage enters the formulation. The augmentation energy accounts for the contribution from vacuum. Finally, we introduce a so-called dissipation function, by means of which irreversible polarization due to domain switching is accounted for. Numerical results computed with the plate model are compared to results using a three-dimensional formulation.