A tetragonal switching model for ferroelectric materials

In this work a tetragonal material model for ferroelectric materials including a microscopically motivated switching criterion is presented. The resulting formulation is able to describe ferroelectric switching effects on a microscopic scale under consideration of the natural tetragonal structure of the ferroelectric material. (© 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 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 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)     

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。     

Modelling of cracks on different scales in ferroelectric materials

In this work we study contributions to the effective fracture toughness of ferroelectric materials arising from effects on macroscopic and mesoscopic scales of the system. On the macroscopic scale, the crack in a ferroelectric material is modeled taking into account an extended theory of stresses at interfaces in dielectric solids [1-3]. We predict several new effects, such as the “poling effect”, “collinear effect” and the coupling of a Mode-II shear loading and the Mode-I SIF. Further, on the mesoscopic scale, we study the influence of polarization switching limited to the fracture process zone (small scale switching) on the fracture toughness. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

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 phenomenological constitutive model for magnetostrictive materials and ferroelectric ceramics

The contribution is concerned with a thermodynamic consistent constitutive model for magnetostrictive materials and ferroelectric ceramics. It captures the nonlinear phenomenological behavior which is described by hysteresis effects. Magnetostrictive alloys and ferroelectric ceramics belong to the multifunctional materials. In recent years these materials have become widely–used in actor and sensor applications. They characterize an inherent coupling between deformation and magnetic or electric field. Due to the similarities of the coupled differential equations a uniform approach is applied for both phenomena. The presented three–dimensional material model is thermodynamically motivated. It is based on the definition of a specific free energy function and a switching criterion. Furthermore an additive split of strain and the magnetic or electric field in a reversible and an irreversible part is suggested. The irreversible quantities serve as internal variables, which is analog to plasticity theory. A one–to–one–relation between the two internal variables provides conservation of volume for the irreversible strains. The presented material model can approximate the ferromagnetic or ferroelectric hysteresis curve and the related butterfly hysteresis. Furthermore an extended approach for ferrimagnetic behavior, which occurs in magnetostrictive materials, is presented. Some numerical simulations demonstrate the capability of the presented model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fracture toughness investigations of ferroelectric materials considering effects on macro- and micro-scale

In the present work we study toughness variation of ferroelectric materials (PZT-5H) considering different scales for different poling and loading conditions. On the macro-scale we apply an extended theory of stresses at interfaces in dielectric solids. Further, on the micro-scale, nonlinear effects are introduced by applying the small scale switching approximation. The analysis is done considering the full anisotropy and electromechanical coupling of the material. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase field modeling of domain evolution in ferroelectric materials in the context of size effects

Phase field modeling provides an efficient tool for the study of domain evolution in ferroelectric materials. Such models naturally introduce an inner length scale which represents the width of the interfaces between domains (domain walls). This inner length scale is of the order of a few unit cells, i.e. about 0.8 nm–2 nm. The focus of this contribution is on size effects in a) the switching behavior of ferroelectric thin films and b) the microstructure evolution in ferroelectric nanodots. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of microcracks in ferroelectric materials by application of a grain-boundary-motivated cohesive law

Ferroelectric materials exhibit a huge potential for engineering applications – ranging from electrical actuators (inverse piezoelectric effect) to sensor technology (direct piezoelectric effect). To give an example, lead zirconate titanate (PZT) is a typical perovskite ion crystal possessing ferroelectric properties. In this contribution, we are particularly interested in the modelling of microcracking effects in ferroelectric materials. In view of Finite-Element-based simulations, the geometry of a natural grain structure, as observed on the so-called micro-level, is represented by an appropriate mesh. While the response on the grains themselves is approximated by coupled continuum elements, grain boundaries are numerically incorporated via so-called cohesive-type elements. For the sake of simplicity, switching effects in the bulk material will be neglected. The behaviour of the grain boundaries is modelled by means of cohesive-type laws. Identifying grain boundaries as potential failure zones leading to microcracking, cohesive-type elements consequently offer a great potential for numerical simulations. As an advantage, in the case of failure they do not a priori result in ill-conditioned systems of equations as compared with the application of standard continuum elements to localised deformations. Finally, representative constitutive relations for both the bulk material and the grain boundaries, enable two-dimensional studies of low-cycle-fatigue motivated benchmark boundary value problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

由上下电极覆盖的铁电薄膜中最小180度电畴尺寸

在较低的电压下,铁电畴发生反转的性能对于研发高密度铁电存储器是至关重要的．为了实现高密度存储,铁电畴必须做得愈小愈好．然而,当外加电场撤去后,很小的铁电畴是不稳定的,会发生缩小,直致消失,导致存储的信息消失．为解决此问题,发展了一种通用的方法用于决定避免反向反转的铁电畴的尺寸．做为一个例子,确定了由上下电极覆盖的铁电薄膜中最小180度电畴尺寸．该研究可以用于许多相似的问题．     

Toughening effect of ferroelectric ceramics induced by domain switching and dislocations

The multi-scale analysis of fracture toughness of ferroelectric ceramics under complicate mechanical–electrical coupling effect is carried out in this paper. The generalized stress intensity factor (SIF) arising from spontaneous strains and polarization transformation in switching domain zones is accurately obtained by using an extended Eshelby theory. Taking BaTiO₃ ferroelectric ceramic for example, it is discovered that the crack propagation can be induced by domain switching arising from negative electrical field when the crack surface is parallel to the isotropic plane, and the obtained critical electric displacement intensity factor (EDIF) approximates closely to that obtained by the Green’s function method. Additionally, as pinning dislocations and slip dislocations can strongly influence properties of ferroelectric devices and induce the property degradation, it is necessary to investigate the dislocation toughening effects on fatigue and fracture mechanisms. The results show that the dislocation shielding and anti-shielding effects on mode II SIF, mode I SIF and EDIF are obviously different when a dislocation locates at a position near the crack tip. Through the calculation of the critical applied EDIF for crack propagation by using mechanical energy release rate (MERR) theory, it is discovered that the slip angles obviously influence fracture toughness, and the mode II SIF arising from dislocation has little influence on fracture toughness, however, the mode I SIF and EDIF arising from dislocation have great influences on fracture toughness.     

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)     

Experimental and Numerical Investigation of Nonlinear Piezoelectric Effects Including Minor Hystereses

Piezoelectric ceramics are often used in active structures for shape and vibration control. Since the operation range is not limited to small signals the nonlinear behaviour of the actuator under high electric loads has to be known. There are several approaches in literature to model the hysteretic effects, each having its assets and drawbacks. When a model is able to reproduce the minor loops of the strain - electric field hysteresis, it often lacks the consideration of stress dependence which is fundamental for actuators attached to elastic structures. On the other hand constitutive models which take into account all ferroelectric and ferroelastic effects are not capable of representing the minor hystereses in acceptable calculation times. In this work a phenomenological constitutive model is verified using the experimental data of an active plate structure. Therefore, the ceramic is characterised under mechanically unconstrained conditions and afterwards attached onto a steel plate. The bonding to the substructure leads to a mechanical stress depending on the actuation state of the ceramic. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational characterization of magneto-electric composites: the role of ferroelectric pre-polarization

The interaction between electric and magnetic fields enables smart devices which can find applications in sensor technology and data storage. Materials showing magneto-electric (ME) coupling combine different ferroic characteristics. In the present contribution we focus on composites, which combine ferroelectric and ferromagnetic phases due to strain couplings, such that they generate a strain-induced ME coupling. We derive a two-scale homogenization approach for the determination of effective properties in consideration of microscopic morphologies. Furthermore, we demonstrate the strong influence of ferroelectric polarization states on the ME-coefficient by modeling the switching of remanent polarizations on the microscale. (© 2015 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)     

Damage model for ferroelectric materials and its applications to multilayer actuators

In this paper a damage model for ferroelectric materials is presented. It is implemented in terms of a user element in the commercial FEM-code Abaqus. The model is based on micromechanical considerations of domain switching and its interaction with microcrack growth and coalescence. Finite element analysis of a multilayer actuator is performed, showing principal stresses leading to crack initiation and damage of the actuator. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)