Laminate-based modelling of microstructure and switching in ferroelectrics

The purpose of the work is the thermodynamics-based modelling of the polarisation and the deformation microstructure in the ferroelectric single crystal with the help of a laminate-based approach. The incremental variational-based rate-dependent macroscopic model for dissipative ferroelectric material [1] and the laminate-based microscopic model [2] established in the literature are taken as basis and shall be further extended to a single crystal laminate structure dependent on the loading frequency based on the coupled electromechanical framework taking the effect of polarisation into account. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A laminate-based framework for switching and microstructure evolution in polycrystalline ferroelectrics

This contribution deals with the numerical modelling of polycrystalline ferroelectric materials considering a sequential laminate-based approach established for tetragonal single-crystal ferroelectrics. The particular model [1] is considered and extended to predict the material behaviour of poly-crystal tetragonal ferroelectric ceramics. The derived laminate-based model is implemented in a finite element environment to simulate the time-dependent domain evolution and switching response of a bulk polycrystalline ferroelectric ceramic. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A thermodynamic consistent model for an assumed transversely isotropic piezoceramic with nonlinear ferroelectric hysteresis

A characteristic feature of ferroelectric crystals is the appearance of a spontaneous polarisation, where its direction can be reversed by an applied electric field. This quantity, that has a maximum value at high electric‐fields, depends on the loading history of the material. In this paper we discuss a thermodynamic consistent phenomenological model for an assumed transversely isotropic ferroelectric crystal, where the history dependency is modelled by internal variables. The anisotropic behaviour is governed by isotropic tensor functions, depending on a finite set of invariants, that satisfy automatically the symmetry relationships of the considered body. The main goal of this investigation is to capture some characteristics of nonlinear ferroelectrica, such as the polarisation‐electric‐field and the strain‐electric‐field (butterfly) hysteresis loops. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the evolution of microstructures in finite plasticity

Kochmann and Hackl introduced in [1] a micromechanical model for finite single crystal plasticity. Based on thermodynamic variational principles this model leads to a non-convex variational problem. Employing the Lagrange functional, an incremental strategy was outlined to model the time-continuous evolution of a first order laminate microstructure. Although this model provides interesting results on the material point level, due to the global minimization in the evolution equations, the calculation time and numerical instabilities may cause problems when applying this model to macroscopic specimens. In order to avoid these problems, a smooth transition zone between the laminates is introduced to avoid global minimization, which makes the numerical calculations cumbersome compared to the model in [1]. By introducing a smooth viscous transition zone, the dissipation potential and its numerical treatment have to be adapted. We obtain rate-dependent time-evolution equations for the internal variables based on variational techniques and show as an example single slip shear. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanical model for the transformation induced plasticity in polycrystalline steels

Due to the effect of transformation induced plasticity (TRIP) , TRIP-steels are very promising materials, e.g. for the automobile industry. The material behavior is characterized by very complex inner processes, namely phase transformation coupled with plastic deformation and kinematic hardening. We establish a micromechanical model which uses the volume fractions of the single phases, the plastic strain and the hardening parameter in every grain of the polycrystalline material as internal variables. Furthermore, we apply the Principle of the Minimum of the Dissipation Potential to derive the associated evolution equations. The use of a coupled dissipation functional and a combined Voigt/Reuss bound directly results in coupled evolution equations for the internal variables and in one combined yield function. Additionally, we show numerical results which prove our model's ability to give a first prediction of the TRIP-steels' complex material behavior. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The electrocaloric effect in ferroelectrics: nonlinear modeling and simulation

In this paper, the theoretical background of a physically based constitutive model is presented. In addition to the nonlinear ferroelectric behavior, the model considers the nonlinear coupling of thermal and electromechanical fields. Results are presented in terms of a simple analytical solution for a single domain configuration. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Laminate Deformation Microstructure in Finite Gradient Crystal Plasticity

The macroscopic mechanical behavior of many materials crucially depends on the formation and evolution of their microstructure. In this work, we consider the formation and evolution of laminate deformation microstructure in plasticity. Inspired by work on the variational modeling of phase transformation [5] and building on related work on multislip gradient crystal plasticity [9], we present a new finite strain model for the formation and evolution of laminate deformation microstructure in double slip gradient crystal plasticity. Basic ingredients of our model are a nonconvex hardening potential and two gradient terms accounting for geometrically necessary dislocations (GNDs) by use of the dislocation density tensor and regularizing the sharp interfaces between different kinematically coherent plastic slip states. The plastic evolution is described by means of a nonsmooth dissipation potential for which we propose a new regularization. We formulate a continuous gradient-extended rate-variational framework and discretize it in time to obtain an incremental-variational formulation. Discretization in space yields a finite element formulation which is used to demonstrate the capability of our model to predict the formation and evolution of laminate deformation microstructure in f.c.c. Copper with two active slip systems in the same slip plane. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Variational Homogenization Approach to Electromechanical Hystereses Caused by Domain Wall Movement

In recent years, increasing interest in so-called smart materials such as ferroelectric polymers and ceramics has been shown. Those materials are used in various actuators, sensors, and also in medical devices. In this paper, we outline a micro-macro approach to the modeling of macroscopic hystereses which directly takes into account the microstructural evolution of electrically poled domains. To this end, an incremental variational formulation for a gradient-type phase field model is developed and exploited for the simulation of electromechanically coupled problems. The formulation determines the hysteretic response of the material in terms of an energy-enthalpy and a dissipation function which both depend on the microscopic remanent polarization treated as an order parameter. The gradient-type balance law for the phase field can be considered as a generalization of Biot's equation for standard dissipative materials and may be related to the classical Ginzburg-Landau equation. Furthermore, the variational formulation serves as natural starting point for a compact and symmetric finite element implementation of the coupled micromechanical problem covering the displacement, the electric potential, and the microscopic polarization vector. For this three-field scenario we develop a variational-based homogenization method which determines the overall macroscopic hysteretic properties of a polycrystalline aggregate. The proposed computational method can be used as a numerical laboratory for the improvement of microstructural properties. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlocal damage-viscoplastic model for high temperature creep of single crystal superalloys

A new nonlocal damage-viscoplastic model for high temperature creep of single crystal superalloys is developed. It is based on the variational formulation consisting of free energy, plastic and damage dissipation potentials. Evolution equations for plastic strain and damage variables are derived from the minimum principle for dissipation potentials [1]. The model is capable of describing different stages of creep in a unified way. The evolution of dislocation densities of gamma and gamma prime phases in superalloys incorporates plastic deformation. It results in the time-dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage variables. Herein the nonlocal one is considered as numerical treatment to remove mesh-dependence. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A condensed microelectromechanical constitutive and damage model for tetragonal ferroelectrics

A condensed model for ferroelectric solids with tetragonal unit cells is presented. The approach is microelectromechanically and physically motivated, considering discrete switching processes on the level of unit cells and quasi-continuous evolution of inelastic fields on the domain wall level. To calculate multiple grain interactions an interaction tensor is introduced. Hysteresis loops are simulated for pure electric and electromechanical loading, demonstrating e.g. the influence of a compressive preload on the poling process and interaction between statistically arranged crystallits. The residual stresses and the corresponding principle stresses are used to simulate fatigue damage in ferroelectric materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulation and Numerical Implementation of Ferroelectrics at Large Strains

In recent years, an increasing interest has been shown in functional materials such as ferroelectric polymers. For such materials, viscous effects and electric polarizations cause hysteresis phenomena accompanied with possibly large remanent strains and rotations. Ferroelectric polymers have many attractive characteristics. They are light, inexpensive, fracture tolerant, and pliable. Furthermore, they can be manufactured into almost any conceivable shape and their properties can be tailored to suit a broad range of requirements. In this work, continuous and discrete variational formulations are exploited for the treatment of the non-linear dissipative response of ferroelectric polymers under electrical loading. The point of departure is a general internal variable formulation that determines the hysteretic response of this class of materials in terms of an energy storage and a rate-dependent dissipation function. The ferroelectric constitutive assumptions, which account for specific problems arising in the geometric nonlinear setting, are discussed. With regard to the choice of the internal variables, a critical factor is the kinematic assumptions. Here, we propose the multiplicative decomposition of the local deformation gradient into reversible and remanent parts, where the latter is characterized by a metric tensor. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromechanically based stress and strain-rate flow potentials for anisotropic polycrystals

The dissipation of rigid-viscoplastic material behaviour is related to the stress flow potential and its Legendre-Fenchel conjugated strain-rate flow potential. In this paper micromechanically based approximation of these flow potentials, expressed in terms of the deviatoric stresses and strain-rates, is derived from texture measurements. The dual potentials for single face-centered cubic crystals are approximated in terms of Fourier expansion with tensorial texture coefficients. The macroscopic flow stress and strain-rate potentials for an aggregate of polycrystals can be deduced from those of single crystal under the assumption of either homogeneous strain-rate, or homogeneous stress field from the codf measurements. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Point defect interaction in tranversally isotropic ferroelectrics

Functional materials, especially ferroelectrics are used in many devices like actuators, sensors and electronic devices. Due to high amounts of mechanical and electrical load cycles, fatigue phenomena may occur. This so called electric fatigue causes a decrease of the electromechanical coupling capability. It is assumed, that the ability to switch polarisation states, which is the reason for the ferroelectric effect, is decreased in the presence of point defects. These defects are ionic and electronic charge carriers, which can interact with each other, with microstructural elements in the bulk and with interfaces. Accumulation of defects can primarily lead to degradation, because of the loss of polarisation switchability. The interaction of defects in the bulk is simulated to get a better understanding of the defect accumulation processes. A model based on configurational forces can be used to obtain thermodynamic consistent kinetic laws. The material used is transversally isotropic and modelled with linear electromechanical coupling. The focus is on the influence of this material anisotropy on the defect interaction. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational method of deriving the equations of the non-linear mechanics of liquid crystals

The non-linear equations of the dynamics of liquid crystals [1], derived previously by the Poisson brackets method, are derived from the Hamilton-Ostrogradskii variational principle. The variational problem of an unconditional extremum of the action functional in Lagrange variables is investigated. The difference between the volume densities of the kinetic and free energy of the liquid crystal is used as the Lagrangian. It is shown that the variational equations obtained are equivalent to the differential laws of conservation of momentum and the kinetic moment of the liquid crystal in Euler variables, while the Ericksen stress tensor and the molecular field are defined in terms of the derivatives of the free energy.     

Time-continuous modeling of microstructures in single-slip finite elasto-plasticity based on energy relaxation

The non-(quasi)convexity of the free energy of single crystals in finite elasto-plasticity gives rise to the formation of fine-scale fluctuations of the displacement field, which can be interpreted as material microstructures. We outline a numerical procedure to model the origin and subsequent evolution of laminate microstructures. This approach incrementally solves the evolution equations by employing relaxed energy potentials. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanically motivated model for ferroelectrics combined with polygonal finite elements

Piezoelectric materials are one of the most prominent smart materials due to their strong electromechanical coupling behaviour. Ferroelectric ceramics behave like piezoelectric materials under low electrical and mechanical loads, but exhibit pronounced nonlinear response at higher loads due to microscopic domain switching. Modern smart devices consist of complex geometries that may force the ferroelectrics employed within them to experience higher fields than they were originally designed for, so that the material responds within its nonlinear region. Hence, models predicting the nonlinear effects of ferroelectrics under complex loading cases are important from the design point of view. Within standard finite element models dealing with electromechanical problems, each grain may be subdiscretized by several finite elements. This problem can be approximated or rather overcome by a polygonal finite element method, where each grain is modelled by solely one single finite element. In this contribution, a micromechanically motivated switching model for ferroelectric ceramics, as based on volume fraction concepts, is combined with polygonal finite element approach. Related representative numerical examples allow to further study and understand the nonlinear response of this material under complex loading cases. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On domain switching in deformable ferroelectrics, seen as continua with microstructure

We obtain a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by a spontaneous polarization. This is accomplished by assuming the body as comprised of a continuum with vectorial microstructure: in each point of the body therefore a gross and a fine structure are superposed, the gross structure representing a non linear polarizable elastic body and the vectorial fine structure describing the spontaneous polarization.¶Among the distinctive features of ferroelectric materials, the most interesting is represented by the organization of spontaneous polarization into a domain structure, which minimizes electrostatic energy and which can be modified by the application of electric and deformation fields. This process, called "polarization reversal" or "domain switching", is associated with various hysteresis loops, the most typical being those between spontaneous polarization and electric field (dielectric hysteresis), and between strain and electric field ("butterfly" loop).¶From the balance laws of continua with vectorial microstructure and dissipation inequality we arrive at the evolution equation for the spontaneous polarization which allows for both inertial and dissipative terms and describes domain switching. We postulate a simple interaction mechanism between the spontaneous polarization and the pair electric field, deformation and arrive at, in the quasi-static case, to a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics.¶For linearized kinematics we study, in the one-dimensional case, stable relative minimizers and give a simple justification for dielectric hysteresis and butterfly loops: under the hypothesis that the domain walls are sharp interfaces, the solutions we find explain the banded twin domains morphology which is typical of many ferroelectrics.     

A coupled phase-field model for ductile fracture in crystal plasticity

The objective of this work is to present a simplified, nonetheless representative first stage of a phenomenological model to predict the crack evolution of ductile fracture in single crystals. The proposed numerical approach is carried out by merging a conventional well- stablished elasto-plastic crystal plasticity model and a well-known phase-field model (PFM) modified to predict ductile fracture. A two-dymensional initial boundary-value problem of ductile fracture is introduced considering a single crystal Nickel-base superalloy material. the model is implemented into the finite element context subjected to a one-dimensional tension test (displacement-controlled). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromorphic continuum model for electromagnetoelastic solids

The microcontinuum theory of electroelasticity is considered for polarizable dielectrics on the basis of dipole and quadrupole densities as microfields. Electromagnetic contributions to force, couple, and power are derived, and their correspondence with quantities evaluated in terms of macroscopic polarization and magnetization is examined. A constitutive model that accounts for dissipation is proposed via internal variables satisfying suitable evolution equations. This approach reveals different roles of polarization and strain measures in dissipative processes. The link between the spin inertia tensor and the pair of dipole and quadrupole per unit mass is exploited to derive a nonlinear system of governing equations for a reduced set of variables. The special cases of microstretch and micropolar continua are discussed.