Constitutive model of ferroelectric composites with a viscoelastic and dielectric relaxation matrix I

Based on micromechanics and Laplace transformation, a constitutive model of ferroelectric composites with a linear elastic and linear dielectric matrix is developed and extended to the ferroelectric composites with a viscoelastic and dielectric relaxation matrix. Thus, a constitutive model for ferroelectric composites with a viscoelastic and dielectric relaxation matrix has been set up Project supported by the National Natural Science Foundation of China (Grant No. 19891180).     

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)     

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)     

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)     

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)     

Shape control of piezolaminated thin-walled structures using nonlinear piezoelectric actuators

In the present paper a 2D-shell finite element model is proposed to carry out static analysis of piezolaminated composite shells by incorporating nonlinear constitutive relations in order to describe the electromechanical coupling under strong electric fields. The present shell element has 5 mechanical DOFs and 3 electrical DOFs per node. The developed composite piezolaminated shell element is employed to study the static behavior and shaping of spherical antenna reflector laminated with piezo-patches under both weak and strong electric fields. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Peridynamics model for flexoelectricity and damage

A flexoelectric peridynamic (PD) theory is proposed. In the PD framework, the formulation introduces a nanoscale flexoelectric coupling that entails non-uniform strain in centrosymmetric dielectrics. This potentially enables PD modeling of a large class of phenomena in solid dielectrics involving cracks, discontinuities etc. wherein large strain gradients are present and the classical electromechanical theory based on partial differential equations do not directly apply. PD electromechanical equations, derived from Hamilton's principle, satisfy the global balance laws. Linear PD constitutive equations reflect the electromechanical coupling effect, with the mechanical force state affected by the polarization state and the electrical force state in turn by the displacement state. An analytical solution to the PD electromechanical equations is presented for the static case when a point mechanical force and a point electric force act in an infinite 3D solid dielectric. A parametric study on how different length scales influence the response is undertaken. In addition, the model is extended to incorporate damage using phase field – an order parameter, supplemented with a PD bond breaking criterion to study flexoelectric effects in damage and fracture problems. To demonstrate the performance of our proposal, we first simulate, considering small flexoelectricity effect and no damage, an externally pressured 2D flexoelectric disk subjected to a potential difference between the inner and outer surfaces and compare the results with existing solutions in the literature. Next, we simulate a plate with a central pre-crack under tension considering damage and flexoelectricity effects, and study the effect of various constitutive parameters on the damage evolution. We also furnish a classical derivation of phase field based flexoelectricity in Appendix I.     

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)     

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 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 material model of nonlinear fractional viscoelasticity

Multiscale methods are frequently used in the design process of textile reinforced composites. In addition to the models for the local material structure it is necessary to formulate appropriate material models for the constituents. While experiments have shown that the reinforcing fibers can be assumed as linear elastic, the material behavior of the polymer matrix shows certain nonlinearities. These effects are mainly due to strain rate dependent material behavior. Fractional order models have been found to be appropriate to model this behavior. Based on experimental observations of Polypropylene a one-dimensional nonlinear fractional viscoelastic material model has been formulated. Its parameters can be determined from uniaxial, monotonic tensile tests at different strain rates, relaxation experiments and deformation controlled processes with intermediate holding times at different load levels. The presence of a process dependent function for the viscosity leads to constitutive equations which form nonlinear fractional differential equations. Since no analytical solution can be derived for these equations, a numerical handling has been developed. After all, the stress-strain curves obtained from a numerical analysis are compared to experimental results. (© 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)     

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)     

惯性往复振动设备的机电耦合模型和瞬态过程分析

采用建立离散系统动力学方程的矩阵法,建立了双旋转偏重激振的惯性往复振动设备的动力学方程,该动力学方程与两驱动电机的状态方程一起构成了惯性往复振动设备机电耦合的数学模型．通过分别对不同阻尼、不同电机功率时数学模型的数值仿真和对仿真结果分析,揭示了惯性振动设备瞬态过程的实质,提出了减小振动体瞬态振幅和缩短瞬态过程的新方法．为该类设备瞬态过程的智能控制和工程设计提供了可靠的数学模型．     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Point defect interaction in tranversally isotropic ferroelectrics

Functional materials, especially ferroelectrics are used in many devices like actuators, sensors and electronic devices. Due to high amounts of mechanical and electrical load cycles, fatigue phenomena may occur. This so called electric fatigue causes a decrease of the electromechanical coupling capability. It is assumed, that the ability to switch polarisation states, which is the reason for the ferroelectric effect, is decreased in the presence of point defects. These defects are ionic and electronic charge carriers, which can interact with each other, with microstructural elements in the bulk and with interfaces. Accumulation of defects can primarily lead to degradation, because of the loss of polarisation switchability. The interaction of defects in the bulk is simulated to get a better understanding of the defect accumulation processes. A model based on configurational forces can be used to obtain thermodynamic consistent kinetic laws. The material used is transversally isotropic and modelled with linear electromechanical coupling. The focus is on the influence of this material anisotropy on the defect interaction. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A solid-shell finite element for fibre reinforced composites

Fibre reinforced composites consisting of several layers, each of which is composed of a woven fabric embedded in a matrix material, are investigated in this paper. Such materials are characterized by a complex anisotropic behavior, which necessitates a fully three-dimensional formulation of the constitutive equations. On the other hand, they are frequently used in thin shell-like applications. In order to account for the three-dimensional material law while still providing the suitable shape for thin structures, a solid-shell finite element for fibre composite materials is presented herein. Locking phenomena are treated by both the enhanced assumed strain (EAS) concept and the assumed natural strain concept (ANS). Using reduced integration together with hourglass stabilization leads to high computational efficiency. The anisotropic constitutive behavior of the composites is reflected by a micromechanically motivated continuum model, which –together with the solid-shell formulation– allows for an accurate representation of the through-the-thickness stress distribution even for thin structures. (© 2012 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)     

Numerical Modeling Aspects of Dielectric Elastomer Actuators

In modern actuator technology dielectric elastomers are considered as new materials to realize smart actuators which are known as dielectric elastomer actuators (DEAs). In comparison to piezoceramics actuators, DEAs offer the possibility to achieve large deformations with low actuation forces. This property motivates the implementation as artificial muscles since the deformation-force behavior is similar. Other application fields are pumps, deformable surfaces in aerospace, robotics and haptic feedback. The present work introduces the fundamental concepts to describe the electromechanical coupling in the concept of continuum mechanics for finite deformations. As a benchmark a 3D sandwich actuator setup is taken into account to analyze the mechanical compression stability of the elastomer structure, see [1, 2]. This structure is also considered to study the influence of inhomogeneities in the deformation behavior. For this purpose piezoceramic and air inclusions are considered in the finite element mesh. As a last numerical example an elastomer tube with three pairs of electrodes is simulated numerically to motivate the use of dielectric elastomers as peristaltic pumps. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

无粘性土的应力矢量本构模型(Ⅱ)——应用

将史宏彦等提出的无粘性土的应力矢量本构模型应用于分析多种复杂应力路径下材料的变形问题。结果表明，此模型不仅能够很好地反映无粘性土的应力应变非线形、硬化性、剪缩剪胀性、与应力路径的相关性、主应力与主应变增量方向之间的非共轴性以及球偏应力与变形的耦合性等主要变形特性，而且也能够同时考虑主应力轴的旋转和中主应力对土的变形及强度的影响。模型预测结果与试验结果之间的良好吻合表明了该模型的广泛适用性。