A Constitutive Model for Magnetostrictive Materials - Theory and Finite Element Implementation

A 3D macroscopic constitutive law for hysteresis effects in magnetostrictive materials is presented and a finite element implementation is provided. The novel aspect of the thermodynamically consistent model is an additive decomposition of the magnetic and the strain field in a reversible and an irreversible part. Employing the irreversible magnetic field is advantageous for a finite element implementation, where the displacements and magnetic scalar potential are the nodal degrees of freedom. To consider the correlation between the irreversible magnetic field and the irreversible strains a one-to-one relation is assumed. The irreversible magnetic field determines as internal variable the movement of the center of a switching surface. This controls the motion of the domain walls during the magnetization process. The evolution of the internal variables is derived from the magnetic enthalpy function by the postulate of maximum dissipation, where the switching surface serves as constraint. The evolution equations are integrated using the backward Euler implicit integration scheme. The constitutive model is implemented in a 3D hexahedral element which provides an algorithmic consistent tangent stiffness matrix. A numerical example demonstrates the capability of the proposed model to reproduce the ferromagnetic hysteresis loops of a Terfenol-D sample. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A 1D constitutive law for hysteresis effects in piezoceramics

This paper is concerned with a macroscopic constitutive law for domain switching effects, which occur in piezoelectric ceramics. The thermodynamical framework of the law is based on two scalar valued functions: the Helmholtz free energy and a switching surface. In common usage, the remanent polarization and the remanent strain are employed as internal variables. The novel aspect of the present work is to introduce an irreversible electric field, which serves besides the irreversible strain as internal variable. The irreversible electric field has only theoretical meaning, but leads to advantages within the finite element implementation, where displacement and the electric potential are the nodal degrees of freedoms. A common assumption is a one-to-one relation between the irreversible strain and the polarization. This simplification is not employed in the present paper. To accomplish enough space for the polarization, resulting from an applied electric field, the irreversible strains are additively split and a special hardening function is introduced. This balances the amount of space and the domain switching due to polarization. The constitutive model reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for piezoelectric ceramics and it accounts for the mechanical depolarization effect. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A laminate-based modelling approach for rate-dependent switching in ferroelectric materials

This contribution focuses on the sequential laminate-based modelling approach for the numerical simulation of the complex electromechanical material behaviour of ferroelectric single crystals. The construction of engineered domain configurations by using the method of sequential lamination in order to study the domain evolution and polarisation switching in ferroelectric single crystals has recently been carried out in the works of [1–4]. By fulfilling the kinematic and polarisation compatibility conditions between the domain structures in a crystal, the proposed laminate-based formulation is governed by an energy-enthalpy function and by a dissipation potential. The mixed energy-enthalpy, written in terms of the total strains, electric field and a set of internal variables, here the multi-rank laminate volume fractions, governs the dissipative electromechanical response of the ferroelectric crystal, whereas the rate-dependent dissipation potential formulated in terms of the flux of the internal variables describes the time-dependent evolution of the multi-rank laminate volume fractions, subjected to inequality constraints. The model reproduces experimentally observed hysteresis and butterfly curves, characteristic for single crystal ferroelectric materials, when subjected to homogeneous electromechanical loading conditions. (© 2015 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)     

畴极化转动对多晶铁电材料断裂特性的影响

主要基于细观力学方法揭示了畴极化转动对多晶铁电陶瓷的各向异性断裂特性的平均影响。首先,用Eshelby-Mori-Tanaka理论和统计模型分析了无穷大铁电材料体中一椭球夹杂的内、外电弹性场,得到畴极化转动对电弹性场的平均影响;其次,推导了等效多晶铁电陶瓷中含一钱币状裂纹的裂纹扩展力(能量释放率)G_ext,并用它估计了畴极化转动对多晶铁电陶瓷断裂特性的影响。对BaTiO₃陶瓷中裂纹扩展力的计算结果表明,对多晶铁电材料断裂特性分析必须考虑畴极化转动的影响。计算结果得出了与实验相一致的结论:在受较小的力时,外加电场对裂纹扩展产生较大的影响,而且在某种程度上能促进了裂纹扩展。     

A hybrid element formulation for electromechanical problems

The difficulty in the modeling of ferroelectric materials is the coverage of the complicated interactions between electrical and mechanical quantities on the macroscale, which are caused by switching processes on the microscale. In the present work we present an electric hybrid element formulation where the stresses and the electric fields are derived by constitutive relations as presented in [1]. Therefore the displacements, the electric potential and the electric displacements are approximated by bilinear ansatz functions. Applying a static condensation procedure we obtain a modified finite element formulation governed by the degrees of freedoms associated to the displacements and the electric potential. The anisotropic material behavior is modeled within a coordinate-invariant formulation [6] for an assumed transversely isotropic material [4]. In this context a general return algorithm is applied to compute the remanent quantities at the actual timestep. Resulting hysteresis loops for the ferroelectric ceramics are presented. (© 2006 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)     

Theory and Numerics of Rate-Dependent Incremental Variational Formulations in Ferroelectricity

This paper is concerned with macroscopic continuous and discrete variational formulations for domain switching effects at small strains, which occur in ferroelectric ceramics. The developed new three–dimensional model is thermodynamically–consistent and determined by two scalar–valued functions: the energy storage function (Helmholtz free energy) and the dissipation function, which is in particular rate–dependent. The constitutive model successfully reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for ferroelectric ceramics. The rate–dependent character of the dissipation function allows us also to reproduce the experimentally observed rate dependency of the above mentioned hysteresis phenomena. An important aspect is the numerical implementation of the coupled problem. The discretization of the two–field problem appears, as a consequence of the proposed incremental variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a benchmark problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A condensed approach to model the constitutive behavior of multiphase ferroelectric and multiferroic materials

Ferroelectric as well as ferromagnetic materials are widely used in smart structures and devices as actuators, sensors etc. Most of the developed models, describing the nonlinear behavior, are implemented within the framework of the Finite Element Method. Most investigations, however, are restricted to simple boundary value problems under uniaxial or biaxial loading and their goal is the calculation of hysteresis loops or to determine e.g. electromechanical coupling coefficients. Regarding these circumstances, the so-called condensed method (CM) is introduced to investigate the macroscopic polycrystalline ferroelectric material behavior at a macroscopic material point without any kind of discretization scheme. In the presented paper, the CM is extended towards multiphase ferroelectric material behavior. Moreover, first numerical results of a multiphase ferroelectric material at the morphotropic phase boundary are presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constitutive modeling of ferromagnetic and ferroelectric behaviors and application to multiferroic composites

In this paper, the constitutive modeling of nonlinear multifield behavior as well as the finite element implementation are presented. Nonlinear material models describing the magneto-ferroelectric or electro-ferromagnetic behaviors are presented. Both physically and phenomenologically motivated constitutive models have been developed for the numerical calculation of principally different nonlinear magnetostrictive behaviors. Further, the nonlinear ferroelectric behavior is based on a physically motivated constitutive model. On this basis, the polarization in the ferroelectric and magnetization in the ferromagnetic and magnetostrictive phases, respectively, are simulated and the resulting effects analyzed. Numerical simulations focus on the calculation of magnetoelectric coupling and on the prediction of local domain orientations going along with the poling process, thus supplying information on favorable electric-magnetic loading sequences. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Magnetostrictive actuation of a smart beam with hysteretic material behaviour

Magnetostrictive materials can be used as actuators in smart structures technology. The relation between induced strain and the applied magnetic field is nonlinear and shows hysteretic behaviour. Thus the magnetomechanical coupling coefficient is not constant and should be defined as a function of strain or magnetic field in computations. In this study the hysteresis of a mechanically unconstrained actuator is determined using the Michelson interferometry. The hysteretic behaviour is modelled phenomenologically by a Preisach model. Using these experimental data for the modelling of an active structure with embedded magnetostrictive actuators, the actual coupling coefficient can be determined utilising the Preisach model. With this procedure the actuation strain of an embedded actuator, including the physical nonlinearities, can be calculated using the material characteristics obtained with an unconstrained actuator. For the determination of the actual coupling coefficient a strain- and field-dependent approach is used. For an experimental validation of the method outlined above, a magnetostrictive actuator is characterised experimentally and then applied to a cantilever aluminium beam. Then, the tip displacement of the actuated beam is measured with a laser triangulation sensor and compared with the numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear numerical simulation of ferroelectric-ferromagnetic multifunctional composites

The theoretical background of nonlinear constitutive multifield behavior is presented. Nonlinear material models describing the ferroelectric or ferromagnetic behaviors are presented. Both physically and phenomenologically motivated constitutive models have been developed for the numerical calculation of the nonlinear magnetostrictive and ferroelectric behaviors. On this basis, the polarization in the ferroelectric and magnetization in the ferromagnetic respectively magnetostrictive phases are simulated and the resulting effects analyzed. The developed tools enable the prediction of the electromagnetomechanical properties of smart multiferroic composites and supply useful means for their optimization. Goals are to improve the efficiency of ME coupling and to reduce damage associated with the poling process. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence and regularity of solutions to a system of nonlinear Maxwell equations

We consider the model that has been suggested by Greenberg et al. (Physica D 134 (1999) 362-383) for the ferroelectric behavior of materials. In this model, the usual (linear) Maxwell's equations are supplemented with a constitutive relation in which the electric displacement equals a constant times the electric field plus an internal polarization variable which evolves according to an internal set of nonlinear Maxwell's equations. For such model we provide rigorous proofs of global existence, uniqueness, and regularity of solutions. We also provide some preliminary results on the long-time behavior of solutions. The main difficulties in this study are due to the loss of compactness in the system of Maxwell's equations. These results generalize those of Greenberg et al., where only solutions with TM (transverse magnetic) symmetry were considered.     

On The Formulatio and Numerical Implementation of Dissipative Electro-Mechanics at Large Strains

In the last year an increasing interest in functional materials such as ferroelectric polymers and ceramics has been shown. For those materials viscous effects or electric polarizations cause hysteresis phenomena accompanied with possibly large remanent strains. This paper outlines aspects of the formulation and numerical implementation of dissipative electro-mechanics at large strains. In the first part, we focus on the geometric nature of dissipative electro-mechanics. In a second part, we discuss constitutive assumptions which account for specific problems arising in the geometric nonlinear setting. This concerns the definition of objective energy storage and dissipation functions with suitable symmetries and (weak) convexity properties. With regard to the choice of the internal variables entering these functions, a critical point are kinematic assumptions. Here, we investigate the multiplicative decomposition of the local deformation gradient into reversible and remanent parts as well as the introduction of a remanent metric. In a third part, we summarize details of the constitutive updates as well as the finite element formulations, both on the basis of compact incremental variational principles. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the constitutive behavior of ferroelectric,ferromagnetic and multiferroic materials by using the condensed method (CM)

Due to the multifunctional applicability, smart materials are of particular interest in the field of material modeling. Most of the developed models, describing the nonlinear behavior, are implemented within the framework of the Finite Element Method (FEM). However, most investigations are restricted to simple boundary value problems (BVP) under uniaxial loading and their goal is the calculation of hysteresis loops. Regarding this circumstance, the so-called condensed method (CM) is introduced to investigate the macroscopic polycrystalline ferroelectric material behavior at a global material point without any kind of discretization scheme. In the presented paper, the CM is extended towards ferromagnetic and multiferroic material behavior. Moreover, numerical results for a pure ferromagnetic behavior and a comparison between the magnetoelectric coupling coefficient calculated by the FEM and the CM are presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A reformulation of the Jiles-Atherton hysteresis model for ferromagnetic/ferroelectric components

The original Jiles-Atherton model for modeling hysteresis in ferroelectric media relates the magnetization, or likewise the magnetic polarization, and the magnetic field strength. Since discrete dimensional electromechanical systems are usually modeled using the flux formulation the Jiles-Atherton model is not applicable in its original form. In this contribution a reformulation in terms of the magnetic flux density is presented. (© 2009 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)     

Phase field simulations in ferroelectric materials

The hindering of domain wall movement by defects in ferroelectric materials is closely connected to electric fatigue. A movable domain wall in a ferroelectric material in most cases is modelled as a singular surface which allows the use of configurational forces. In contrast, the present approach treats the polarization as an order parameter, extending the total energy by a phase separation energy and a domain wall energy. The polarization then no longer has a discontinuity at the domain wall but is a continuous vector field (phase field). As an example, a numerical simulation of domain evolution under stress free boundary conditions is presented. (© 2006 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)     

Phase field simulations on the interaction of ferroelectric domains with point defects

One of the suspected micro-mechanical mechanisms causing electric fatigue in ferroelectric materials is the hindering and blocking of domain wall movement. These blocking or pinning phenomena are thought to be due to point defects which interact with domain walls and applied external loads. A phase field model employing the spontaneous polarization as an order parameter is used to simulate the inhomogeneous material behavior. The coupled field equations are solved using the Finite Element Method. The influence of a stationary point defect on a domain wall is shown in a numerical simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)