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)     

An energetic material model for time‐dependent ferroelectric behaviour: existence and uniqueness

We discuss rate‐independent engineering models for the multi‐dimensional behaviour of ferroelectric materials. These models capture the non‐linear and hysteretic behaviour of such materials. We show that these models can be formulated in an energetic framework which is based on the elastic and the electric displacements as reversible variables and on interior, irreversible variables like the remanent polarization. We provide quite general conditions on the constitutive laws which guarantee the existence of a solution. Under more restrictive assumptions we are also able to establish uniqueness results. Copyright © 2006 John Wiley & Sons, Ltd.     

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)     

Aspects of the simulation of the ferroelectric hysteresis

In this paper we present a thermodynamically consistent phenomenological model for an assumed transversely isotropic ferroelectric crystal. Such materials become spontaneously polarized within a certain range of temperature. On the macro scale a remanent polarization and remanent strains, due to the reorientation of the polarization vectors, are observed, if an electric field above the so-called coercive field is applied. The goal of this work is to construct a thermodynamically consistent formulation of the electro-mechanically coupled ceramic on a meso scale, that takes into account the orientation of the assumed transversely isotropic unit cell, see [4]. The anisotropic behavior is governed by isotropic tensor functions, which are formulated in terms of a finite set of invariants, see [3]. Considering a simple model problem we discuss the linearization of discrete weak forms, resulting from the electromechanical boundary value problem. (© 2005 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 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)     

Configurational forces in a multiscale approach to piezoelectric materials

Due to the growing interest in determining the macroscopic material response of inhomogeneous materials, computational methods are becoming increasingly concerned with the application of homogenization techniques. In this work, a two-scale classical homogenization of an electro-mechanically coupled material using a FE²-approach is discussed. We explicitly formulated the homogenized coefficients of the elastic, piezoelectric and dielectric tensors for small strain as well as the homogenized remanent strain and remanent polarization. In the homogenization different representative volume elements (RVEs), which capture the micro-structure of the inhomogeneous material, are used to represent the macroscopic material response. Two different schemes are considered. In the first case, domain wall movement is not allowed, but in the second case the movement of the domain walls is taken into account using thermodynamic considerations. Later this technique is used to determine the macroscopic and microscopic configurational forces on defects [2]. These defect situations include the driving force on a crack tip. The effect of the applied electric field on configurational forces at the crack tip is investigated. (© 2011 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)     

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)     

Computational Homogenization of Piezoelectric Materials using FE2

Due to the growing interest in determining the macroscopic material response of inhomogeneous materials, computational methods are becoming increasingly concerned with the application of homogenization techniques. In this work, two-scale classical (first-order) homogenization of electro-mechanically coupled problems using a FE²-approach is discussed. We explicitly formulated the homogenized coefficients of the elastic, piezoelectric and dielectric tensors for small strain as well as the homogenized remanent strain and remanent polarization. The homogenization of the coupled problem is done using different representative volume elements (RVEs), which capture the microstructure of the inhomogeneous material, to represent the macro material response. Later this technique is used to determine the macroscopic and microscopic configurational forces on certain defects. (© 2010 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)     

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.     

FEM-analysis of a multiferroic nanocomposite – comparison of experiment and numerical simulation

The combination of electric and magnetic materials opens new possibilities in the field of sensor technologies and data storage [1]. These magneto-electric (ME) materials have the property to change a physical ferroic quantity into another, i.e. a magnetic field can change the electric polarization and vice versa. The combination of multiple ferroic characteristics within materials is called multiferroic. Since magneto-electric single-phase materials are rare in nature and typically operate only at very low temperature, they are not favorable in technical applications. However, ME composites, consisting of ferroelectric and ferromagnetic phases, produce a strain-induced magneto-electric product property at room temperature [2]. In these composites, two different effects can be differentiated, the direct and the converse ME effect. The first one describes a polarization which is magnetically caused. In detail, a magnetic field is applied which produces a deformation of the magneto-active phase which is transferred to the electro-active phase and as a consequence this phase exhibits a polarization. Therefore, one can discover a strain-induced polarization. The second effect to observe is a magnetization caused by an electric field. In our contribution, we focus on a (1-3) composite, where cobalt ferrite nanopillars are embedded in a barium titanate matrix, see the experiments described in [3]. In the numerical simulations we compare the changes of the strain-induced inplane polarizations of the ferroelectric matrix with experimental measurements. Furthermore, we analyze the magneto-electric coupling coefficient. (© 2017 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)     

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 thermodynamical consistent model for modeling of ionic electroactive polymers (EAPs) within the framework of the Theory of Porous Media

Ionic electroactive polymers are widely used in many engineering fields. These kind of materials can be stimulated to change their shape and size, see [1]. Since, the material under consideration has a complex multiphasic microstructure, such multiphasic materials are best described by a continuum mechanical approach. Thus, the presented model is based on the Theory of Porous Media (TPM), cf. [2]. In this contribution, we consider the Ionic Polymer Metal Composites (IPMCs). Stimulating by an electrical voltage, a structural deformation will be caused. Responsible for this deformation are the mobile ions. The focus of the presented model is to capture this material behavior, e.g. the distribution of the mobile cations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Treatment and Stability Analysis of Coupled Electro-Mechanics

Dielectric materials such as electro-active polymers (EAPs) belong to the class of functional materials which are used in advanced industrial environments as sensors or actuators and in other innovative fields of research. Driven by Coulomb-type electrostatic forces EAPs are theoretically able to withstand deformations of several hundred percents. However, large actuation fields and different types of instabilities prohibit the ascend of these materials. One distinguishes between global structural instabilities such as buckling and wrinkling of EAP devices, and local material instabilities such as limit- and bifurcation-points in the constitutive response. We outline variational-based stability criteria in finite electro-elastostatics and design algorithms for accompanying stability checks in typical finite element computations. These accompanying stability checks are embedded into a computational homogenization framework to predict the macroscopic overall response and onset of local material instability of particle filled composite materials. Application and validation of the suggested method is demonstrated by a representative model problem. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)