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)     

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)     

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)     

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)     

Theoretical study and numerical simulation of textiles

We propose a new approach for developing continuum models fit to describe the mechanical behavior of textiles. We develop a physically motivated model, based on the properties of the yarns, which can predict and simulate the textile behavior. The approach relies on the selection of a suitable topological model for the patch of the textile, coupled with constitutive models for the yarn behavior. The textile structural configuration is related to the deformation through an energy functional, which depends on both the macroscopic deformation and the distribution of internal nodes. We determine the equilibrium positions of these latter, constrained to an assigned macroscopic deformation. As a result, we derive a macroscopic strain energy function, which reflects the possibly nonlinear character of the yarns as well as the anisotropy induced by the microscopic topological pattern. By means of both analytical estimates and numerical experiments, we show that our model is well suited for both academic test cases and real industrial textiles, with particular emphasis on the tricot textile.     

Linear and nonlinear Finite Element modeling of coupled electro-magneto-mechanical multifield problems

In this paper we present the theoretical background and application of Finite Element algorithms for linear and nonlinear problems of multiple field coupling. They enable the prediction of the electromagnetomechanical behavior of materials and structures and supply useful tools for the optimization of multifunctional composites. First, linear three-field coupling is presented within the context of a Finite Element implementation. Then, a homogenization procedure is discussed. Finally, a micromechanical model for nonlinear ferroelectric constitutive behavior and its numerical realization are outlined. (© 2013 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 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)     

The Cauchy problem in one-dimensional nonlinear viscoelasticity

The initial value problem for a nonlinear hyperbolic Volterra equation which models the motion of an unbounded viscoelastic bar is studied. Under physically motivated assumptions, the existence of a unique, globally defined, classical solution is established provided the initial data are sufficiently smooth and small. Boundedness and asymptotic behavior are also discussed. This analysis is based on energy estimates in conjunction with properties of strongly positive definite kernels.     

A novel class of model constitutive laws in nonlinear elasticity: Construction via Loewner theory

Using a solitonic connection, we show that the class of infinitesimal Bäcklund transformations originally introduced by Loewner in 1952 in a gasodynamic context results in physically interesting nonlinear model constitutive laws. We obtain laws previously used to model a variety of hard and soft nonlinear elastic responses. A natural extension of the latter leads to a novel class of model constitutive laws where the stress and strain are given parametrically in terms of elliptic functions. Such models allow a change in the concavity of the stress-strain law. Such behavior can be observed in the compression of polycrystalline materials or in the unloading regimes of superelastic nickel-titanium.     

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)     

Finite element simulation of the poling process in BaTiO3–CoFe2O4 multiferroic composites

The theoretical background of nonlinear constitutive magneto-ferroelectric behavior as well as the Finite Element implementation are presented. On this basis the polarization in the ferroelectric matrix (BaTiO₃) with embedded dielectric-magnetostrictive particels (CoFe2O₄) is simulated and the resulting effects are analyzed. Numerical simulations focus on the prediction of local crystal orientations and residual stress going along with the poling process, in the future supplying information on favorable electric-magnetic loading sequences. Further, multifield homogenization procedures enable the prediction of the electromagnetomechanical properties of smart multiferroic composites and supply useful means for their optimization. The resulting final state of a poling simulation can be implemented as a starting condition for approximate linear simulations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-posedness Results for Models of Elastomers

Existence and uniqueness of weak solutions are shown for different models of the dynamic behavior of elastomers. The models are based on a nonlinear stress-strain relationship (satisfying a locally Lipschitz and affine domination property) and incorporate hysteretic effects as well. The results provide alternatives to previous theories that required monotonicity assumptions on the nonlinearities. Results with a nonlinear constitutive law and nonlinear internal dynamics are presented for the first time.     

Modeling,Simulation and Parameter Identification for Rate-Dependent Magnetoactive Polymer Response

A finite deformation framework for nonlinear magneto-viscoelasticity is introduced and applied to the constitutive and structural modeling of magnetoactive polymer (MAP) response. In this thermodynamically-consistent formulation the free energy function consists of purely elastic, purely magnetic and coupling contributions, where the rate-dependence is fully attributed to the non-magnetizable matrix material. The model consistently accounts for saturation in the magnetic as well as the magnetostrictive behavior. The identification of material parameters from experimental data is briefly described. Finally, a finite element model for the large strain magneto-mechanical problem is established and tested considering MAP behavior. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constitutive models of ferroelectric composites with a viscoelastic and dielectric relaxation matrix (II)

Experimental analysis of ferroelectric composites with a viscoelastic and dielectric relaxation matrix is carried out, and the electromechanical coupling behavior of the ferroelectric composites is calculated by means of the constitutive model proposed in this paper. Comparisons between the experimental results and the calculations show that the constitutive model can reflect the electromechanical coupling behavior of the ferroelectric composites. The analysis indicates that the effect of viscoelasticity and dielectric relaxation of the matrix on the electromechanical coupling behavior of ferroelectric composites cannot be neglected.     

Least-squares mixed finite elements with applications to anisotropic elasticity and viscoplasticity

The objective of this work is to discuss a least-squares finite element method with applications to physically nonlinear and anisotropic constitutive equations at small strains. The L₂-norm minimization of the residuals of the given first order system of differential equations leads to a functional, which is a two field formulation in the displacements and the stresses, see e.g. Cai & Starke [1]. These functionals provide the foundation for the formulations of the related least-squares mixed finite elements. A main focus of the presentation lies on the extension of plane elasticity to anisotropic or nonlinear material behavior. In this context transversely isotropic elasticity and viscoplasticity is considered. Finally a numerical example is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the thermodynamics of fluids defined by implicit constitutive relations

In this paper, we develop a thermodynamically consistent theory for describing the response of nonlinear viscous fluids whose constitutive equations are of the form f (T, D) = 0. We show that such constitutive equations which include classical constitutive equations wherein the stress is expressed explicitly in terms of the kinematical quantities, provide a rich class of physically meaningful fluid response functions which allows us to describe a wider range of material behavior, including that of a general class of incompressible fluids, incompressible fluids with pressure dependent viscosity, and Bingham (or pseudoplastic) materials.     

Computational characterization of ferroelectric solids with a 3D Preisach model based on an orientation distribution function

Ferroelectric or ferromagnetic materials show an interaction between mechanical deformations and polarization or magnetization. A few multiferroic materials possess both ferroic properties and exhibit a magneto-electric (ME) coupling. These ME properties can be achieved in two-phase composites, which combine ferroelectric and ferromagnetic characteristics. To predict a realistic material behavior and a more precise ME coefficient, the application of suitable material models which describe the nonlinear hysteretic behavior is of particular importance. In the present contribution we focus on the characterization of a nonlinear ferroelectric material behavior, in terms of a 3D Preisach model based on an orientation distribution function. (© 2016 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 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)