<title>Effective properties of piezoelectric polycrystals</title>

A microstructural model for homogenisation and cracking of piezoelectric polycrystals

“…It is possible to observe how the scattering of the apparent properties is reduced as the number of grains in the individual realisations grows, then approaching the effective values of the material properties. The homogenization results appear consistent with the values reported in Ref [75]…”

“…Ref. [75] (c) is the vacuum permittivity constant. Constants' subscripts are given according to Voigt notation.…”

Effective elastic properties for unpoled barium titanate

“…In previous work the focus of FE modelling has been to replicate the material microstructure, 9,10 requiring high quality meshing with many elements. Due to this complexity, analysis has typically been limited to small representative volumes containing only a few grains.…”

“…Further simplification is achieved by modelling in two-dimensions, although results from 2D plane strain or plane stress do not agree with comparable full 3D models. 9,10 The approach adopted in this work was to depreciate the role of microstructure and develop an efficient three-dimensional model. This was based on the assumption that the elastic response is primarily governed by intrinsic atomistic factors.…”

“…From this simple approach various improvements have been sought, first accounting for the inter-crystallite piezoelectric coupling, 6 to self-consistent effective medium methods, 7,8 to recent micro-mechanical finite element (FE) approaches. 9,10 However, the inherent weakness of these schemes is the requirement for single crystal data, which is not easily obtainable for many ferroelectrics, especially solid-solution materials such as PZT.…”