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)     

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)     

Micromechanically motivated finite element model for ferroelectric ceramics

Ferroelectric ceramics exhibit significant coupled electromechanical phenomena that have been widely employed in sensor and actuator applications. In regular finite element models dealing with electromechanical plane problems, each grain needs to be subdiscretized by many triangular or quadrilateral elements for required accuracy. This problem can be overcome by a polygonal finite element approach where each grain is modelled by a single finite element without compromising on the results. In this paper, a polygonal finite element approach has been employed to understand the anisotropic response of the ferroelectric ceramics in their piezoelectric region. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A constitutive model for thermoplastic materials subjected to high strain rates

Reliable prediction of the behaviour of structures made from polymers is a topic under considerable investigation in engineering practice. Especially, if the structure is subjected to dynamic loading, constitutive models considering the mechanical behaviour properly are not available in commercial finite element codes yet. A constitutive model is derived including important phenomena like necking, strain rate dependency, unloading behaviour and damage. In particular, different yield surfaces in compression and tension and strain rate dependent failure, the latter with damage induced erosion, is taken into account. With the present formulation, standard verification tests can be simulated successfully. Also, an elastic damage model can be used to approximate the unloading behaviour of thermoplastics adequately. (© 2005 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)     

Development of an elastic-plastic constitutive model for highly anisotropic materials

Homogenization methods are used to obtain the effective properties of highly anisotropic materials such like textile reinforced composites. The state of the art is the utilization of this method to compute elastic properties. But the consideration of inelastic and anisotropic properties requires the application of more advanced techniques such as the FE²-method. Due to the high numerical effort induced by this approach, this paper presents a new method to evaluate inelastic properties of an heterogeneous elastic-plastic material. The parameters describing the inelastic properties require a modification of the return mapping algorithm which is used for the numerical implementation. Finally, the verification shows the accuracy of the results obtained with this new homogenization method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A constitutive model for granular materials with microstructures using the concept of energy relaxation

In this paper, we present a constitutive model for granular materials exhibiting microstructures using the concept of energy relaxation. Within the framework of Cosserat continuum theory the free energy of the material is enriched with an interaction energy potential taking into account the counter rotations of the particles. The enhanced energy potential fails to be quasiconvex. Energy relaxation theory is employed to compute the relaxed energy which yields all possible displacement and micro-rotations field fluctuations as minimizers. Based on a two-field variational principle the constitutive response of the material is derived. The developed constitutive model is then implemented in a finite element analysis program using the finite element method. Numerical simulations are presented to observe the localized deformation phenomenon in a granular medium. (© 2013 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)     

Non-linear constitutive relations for unstable materials

Non-linear constitutive relations to describe the rheological properties of unstable materials in the one-dimensional case are proposed, where the properties of instability, in particular ageing (i.e., the increase in stiffness and toughness of the material with time), are described by experimentally determined “instability functions”, which depend on the “age” of a given batch of material and, possibly, on invariance of the stress or strain tensors, which affect the rate of physical-chemical processes in the materials. Taking into account the instability of the material, a “fast” time is introduced, measured from the beginning of a short-term test with a material of a given age, and a “slow” time, measured from the instant when the material is produced. An exact solution of the problem of identifying the model, i.e., a determination of the material instability functions from experimental data, is constructed.     

A comparison of two microplane constitutive models for quasi-brittle materials

This article presents a comparison of two microplane constitutive models. The basis of the microplane constitutive models are described and the adopted assumptions for the conception of these models are discussed, with regard to: decomposition of the macroscopic strains into the microplanes, definition of the microplane material laws, including the choice of variables that control the material degradation, and homogenization process to obtain the macroscopic quantities. The differences between the two models, with respect to the employed assumptions, are emphasized and expressions to calculate the macroscopic stresses are presented. The models are then used to describe the behavior of quasi-brittle materials by finite element simulations of uniaxial tension and compression and pure share stress tests. The results of the simulations permit to compare the capability of the models in describing the post critical strain-softening behavior, without numerically induced strain localization.     

A thermodynamically consistent constitutive model for fluid mud

In this contribution, an approach towards a thermodynamically consistent constitutive model for fluid mud is presented. Fluid mud exhibits highly non-Newtonian, thixotropic behaviour. It can be classified as a structured fluid. Typically, its viscosity is modeled using Bingham-type rheological models of different complexity [1, 2]. Here, the three-dimensional non-Newtonian constitutive behaviour will be modeled based on a visco-elasto-plastic model. At the current stage, a Drucker-Prager-like yield function has been formulated. Viscosity is assumed to be a function of shear viscosity. First results show the general ability to represent experimental data. (© 2014 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)     

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 note on the mechanical constitutive equations for materials with memory

Zusammenfassung Die Arbeit behandelt die Einschränkungen, welche sich für die Stoffgleichungen eines Materials mit Gedächtnis daraus ergeben, dass die simultane Rotation von Körper und Bezugssystem die Spannungen unverändert lassen muss. Zwei frühere Methoden zur Gewinnung dieser Einschränkungen wurden vertieft.     

A constitutive model for continuous fiber reinforced polymer plies

In the present paper a constitutive model is reviewed which can be used to predict the non-linear behavior of continuous fiber reinforced laminates with polymeric matrix materials. The constitutive model considers stiffness degradation and plastic strain accumulation at the length scale of the individual plies (laminae). These effects are modeled via two different phenomenological approaches, however, their interaction is considered when the constitutive equations are solved by an implicit integration scheme. To demonstrate the predictive capabilities of the individual model parts, examples are given where the above mentioned effects are decoupled. This way, their impact on the laminate's response can be studied independently. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A constitutive model for a high alloyed cast CrMnNi TRIP-steel

The macroscopic material model, proposed by Papatriantafillou [1], is adopted to describe the rate dependent flow behaviour and the temperature and stress state dependent γ-α′ phase transformation of a newly developed cast TRIP-steel. Simple test of the implemented model revealed that the evolution of the martensite volume fraction is not predicted correctly in case of inelastic deformation and subsequent unloading. Therefore, we present an improved model for the γ-α′ phase transformation and show that these predictions are due to the choice of the corresponding thermodynamical driving force. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the constitutive relations for isotropic and transversely isotropic materials

In this work the strain and stress spaces constitutive relations for isotropic and transversely isotropic softening materials are developed. The loading surface is considered in the strain space and the normality rule; the stress relaxation is proportional to the gradient of the loading surface, is adopted. It is found that the strain space plasticity theory allows us to describe the hardening, perfectly plastic and softening materials more accurately. The validity of the strain space constitutive relation for transversely isotropic materials are confirmed by comparing with the experimental data for fiber reinforced composite materials. Some numerical examples in two and three dimensional elasto-plastic problems for various loading–unloading conditions are presented, and give a very good agreement with the existing results.