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ARTICLE

2D and 3D Multiphysics Voronoi Cells, Based on Radial Basis Functions, for Direct Mesoscale Numerical Simulation (DMNS) of the Switching Phenomena in Ferroelectric Polycrystalline Materials

Peter L. Bishay¹, Satya N. Atluri¹

1 International Collaboratory for Fundamental Studies in Engineering Sciences, University of California, Irvine, CA, USA

Computers, Materials & Continua 2013, 33(1), 19-62. https://doi.org/10.3970/cmc.2013.033.019

Abstract

In this paper, 2D and 3D Multiphysics Voronoi Cells (MVCs) are developed, for the Direct Mesoscale Numerical Simulation (DMNS) of the switching phenomena in ferroelectric polycrystalline materials. These arbitrarily shaped MVCs (arbitrary polygons in 2D, and arbitrary polyhedrons in 3D with each face being an arbitrary polygon) are developed, based on assuming radial basis functions to represent the internal primal variables (mechanical displacements and electric potential), and assuming linear functions to represent the primal variables on the element boundaries. For the 3D case, the linear functions used to represent the primal variables on each of the polygonal surfaces of the polyhedral VCs are the Barycentric Washspress functions. The present 2D MVC is denoted as MVC-RBF, while the 3D MVC is denoted as MVC-RBF-W. Each MVC can represent a single grain or crystallite, with an irregular polygonal shape for the 2D case, and an irregular polyhedral shape for the 3D case. In this work, a nonlinear constitutive model is used to describe the evolution of volume fractions of the constitutive-variants in each grain, as the electric or mechanical loading changes. This constitutive model is based on satisfying a local dissipation inequality in each grain in the polycrystalline that yields the minimum Gibbs free energy in this grain. This requirement should always hold in order to be consistent with the second law of thermodynamics and is used to govern the switching process in each grain in each simulation step. Since the interaction between the grains during the loading cycles has a profound influence on the switching phenomena, it is important to simulate the grains with geometrical shapes that are similar to the real shapes of the grains as seen in the lab experiments. Hence the use of 3D MVCs, which allow for the presence of all the six variants of the constitutive relations, together with the randomly generated crystallographic axes in each grain (or MVC), as done in the present paper, is considered to be the most realistic analytical model that can be used for the direct mesoscale numerical simulation of polycrystalline ferroelectric materials.

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