Return mapping algorithms and consistent tangent operators in ferroelectroelasticity

Abstract

AbstractReturn mapping algorithms for a rather general class of phenomenological rate‐independent models for ferroelectroelastic materials are presented. The fully coupled thermodynamically consistent three‐dimensional constitutive model with two internal variables (remanent polarization vector and remanent strain tensor) proposed by C. M. Landis in 2002 is used for the simulation of electromechanical hysteresis effects in polycrystalline ferroelectric ceramics. Based on the operator splitting methodology, the return mapping algorithm employs the closest point projection scheme to obtain an efficient and robust integration of the constitutive model. The consistent tangent operator is obtained in closed form by linearizing the return mapping algorithm, and is found to be non‐symmetric in the general case due to the dependence of the switching criterion on internal variables. Conditions that provide the symmetry of the consistent tangent matrix are analyzed. The compactness and generality of the received relations are achieved by means of using the thermodynamically based compact notations combining mechanical and electrical values. Both the cases scalar potential finite element (FE) formulation (primary variables: strain and electric field) and vector potential FE formulation (primary variables: strain and electric displacement) are considered. The accuracy and robustness of the algorithms are assessed through numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.