A rate-dependent incremental variational formulation of ferroelectricity

This paper presents a variational-based modeling and computational implementation of the non-linear, rate-dependent response of piezoceramics under electro-mechanical loading. The point of departure is a general internal variable formulation that describes the hysteretic electro-mechanical response of the material as a standard dissipative solid. Consistent with this type of dissipative continua, we develop a variational formulation of the coupled electro-mechanical boundary-value-problem based on incremental potentials for the stresses and the electric displacement. We specify the variational formulation to a model that describes time-dependent, electric polarizations accompanied by remanent strains. It is governed by a dual dissipation function formulated in terms of the internal driving forces. The model reproduces experimentally observed dielectric and butterfly hystereses, which are characteristic for ferroelectric materials. It accounts for the rate-dependency of the hystereses and the macroscopically non-uniform distribution of the polarization in the solid. An important aspect of our treatment is the numerical implementation of the coupled problem. The monolithic discretization of the two-field problem appears, as a consequence of the proposed variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a spectrum of benchmark problems.