Dissipative ferroelectricity at finite strains. Variational principles, constitutive assumptions and algorithms

In recent years increasing interest in functional materials such as ferroelectric polymers and ceramics has been shown. For those materials, electric polarizations and viscous effects cause dissipative phenomena, such as the characteristic dielectric and butterfly hystereses. The deformation of polymers is characterized by large strains and rotations. This work develops a general framework for the formulation and numerical implementation of ferroelectric materials at finite deformations. In particular, continuous and discrete variational principles for the dissipative response of quasi-static finite electro-mechanics are developed, which fully determine the continuous evolution and incremental update problems. These principles recast the framework into a canonically compact structure, that makes the model-inherent symmetries of the coupled problem transparent. In the algorithmic stetting, two-step update schemes are developed, first for the local internal variables and next for the global primary fields, both fully determined by the exploitation of incremental potentials. Specific constitutive assumptions are proposed for finite ferroelectricity. A critical point is the definition of kinematic assumptions in the large-strain context, such as the multiplicative decomposition of the deformation gradient into reversible and remanent parts. The proposed formulation allows to reproduce dielectric and butterfly hystereses together with their rate-dependency and accounts for macroscopically non-uniform distributions of the polarization at finite deformations. The performance of the proposed methods is demonstrated by means of benchmark problems undergoing large deformations.