Asymptotic behavior of thin ferroelectric models

Abstract

It was pointed out in Shaw et al. (2000) that the boundary conditions satisfied by the polarization play an important role in the description of the thin-limit of ferroelectric materials. In this work we confirm the importance of this choice. In the present work, we consider the limiting process as the thickness $h$ of a thin cylinder of $R^3$ goes to 0, and when the polarization satisfies two different boundary conditions. The first type of boundary conditions leads to an in-plane model or $2d-2d$ configuration while the second one leads to an (in-plane)-(out-of-plane) model for the displacement and the polarization namely a $2d-3d$ configuration. Moreover, the thin-limit process in both cases induces a change of the Lamé coefficients in the displacement equation, the coupling coefficient between the displacement and polarization equations, the double wells potential of the polarization together to a new contribution in the equation of the out-of-plane component of the polarization for the $2d-3d$ model. The techniques used rely on a rescaling method that penalizes the out-of-plane variable. Uniform bounds with respect to $h$ are established, compactness techniques are employed, and the limits of the penalized terms are identified.