Laminate-based modelling of single and polycrystalline ferroelectric materials - application to tetragonal barium titanate

The present contribution deals with the development of a laminate-based model designed to study the single and polycrystalline tetragonal ferroelectric material behaviour. Laminate-based models are micromechanically motivated and consider the volume fraction of the distinct ferroelectric variants directly in their formulation. At first, a single crystal laminate-based model is established by considering the average strain and polarisation compatibility conditions. A suitable thermodynamic electric Gibbs energy and a rate-dependent dissipation equation are postulated to capture the dissipative hysteretic material response. The update of the inequality constrained volume fractions is solved by adopting a Fischer-Burmeister-type algorithm in combination with a Newton-Raphson scheme. Following the single crystal formulation, a homogenisation procedure based on random orientation of the individual grains in a polycrystalline aggregate is considered. The material properties and the polarisation switching response of the randomly oriented individual grains are averaged using a finite element framework in order to study the macroscopic polycrystalline behaviour. A parameter fitting procedure based on experimental data for single crystalline response, taken from the literature, is detailed and the material model as well as the algorithmic scheme are verified by solving representative boundary value problems. Moreover, the finite element based simulation results are compared with newly generated experimental hysteresis data for a barium titanate piezoceramic.