A finite element modeling of piezoelectric materials with a superimposed stress

Based on the mechanism of domain switching, a three dimensional nonlinear finite element model for piezoelectric materials subjected to electromechancial loading is developed in this contribution. The finally considered model problem deals with differently oriented grains whereby uni-axial, quasi-static cyclic loading is applied. It is assumed that a crystal orientation switches if the reduction in free energy of the grain exceeds a critical energy barrier. The nonlinearity in the small electromechanical loading range is addresses via a polynomial probability function for domain switching. Hysteresis behavior is discussed taking the influence of a superimposed compression state into account. It is observed that the hysteresis loop flattens under the axial compression but elongates under the transverse compression. Irrespective of how the compression is applied, the remnant polarization and as well as the coercive electric field decrease. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of nonlinear properties of ferroelectric and piezoceramicmaterials

Ferroelectric and piezoelectric materials are becoming a very significant part of smart materials that are used widely as actuators, sensors and most common applications such as vibration control, precision positioning, precision cutting and microelectromechanical systems (MEMS). Piezoceramic materials show nonlinear characteristics when they are under high electromechanical loading. In this study, nonlinear behaviour of tetragonal perovskite type piezoceramic materials is simulated using micromechanical model. In the simulations uni‐axial loading is applied. The calculations which are based on a linear constitutive model, nonlinear domain switching model and a model of probability to switch are performed at each grain. The different domain switching effects (900 or 1800 domain switching for tetragonal perovskite structure) due to energy differences, different probability functions, different statistical random generators and material parameters are analyzed. Finally, simulation results are compared with the data of experiments are giving in literature. (© 2004 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)     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.     

Accceleration of Fatigue Tests of Polymer Composite Materials by Using High-Frequency Loadings

The possibility of using high-frequency loading in fatigue tests of polymer composite materials is discussed. A review of studies on the use of high-frequency loading of organic-, carbon-, and glass-fiber-reinforced plastics is presented. The results obtained are compared with those found in conventional low-frequency loadings. A rig for fatigue tests of rigid materials at loading frequencies to 500 Hz is described, and results for an LM-L1 unidirectional glass-fiber plastic in loadings with frequencies of 17 and 400 Hz are given. These results confirm that it is possible to accelerate the fatigue testing of polymer composite materials by considerably increasing the loading frequency. The necessary condition for using this method is an intense cooling of specimens to prevent them from vibration heating.     

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)     

Modal analysis of piezoelectric bodies with voids. I. Mathematical approaches

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.     

Modal analysis of piezoelectric bodies with voids. II. Finite element simulation

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.     

Application of a hybrid meshless technique for natural frequencies analysis in functionally graded thick hollow cylinder subjected to suddenly thermal loading

A hybrid meshless technique based on composition of meshless local Petrov–Galerkin method (for spatial variables) and Newmark finite difference method (for time domain) is developed for natural frequencies analysis of thick cylinder made of functionally graded materials (FGMs). The FG cylinder is assumed to be under suddenly thermal loading, axisymmetric and plane strain conditions. The dynamic behaviors and time history of displacements are obtained in time domain using Green–Naghdi (GN) theory of coupled thermo-elasticity (without energy dissipation). Using fast Fourier transform (FFT) technique, the displacements are transferred to frequency domain and all natural frequencies are illustrated for various grading patterns of FGMs. The variations of mechanical properties in FG thick hollow cylinder are considered to be in nonlinear volume fraction law through radial direction. The presented hybrid meshless technique furnishes a ground to analyze the effects of various grading patterns of FGMs on natural frequencies, which are obtained employing GN coupled thermo-elasticity governing equations. Also, the frequency history and natural frequencies are illustrated for various grading patterns at several points across thickness of cylinder.     

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)     

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。     

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)     

Elasticity Solutions for Cylindrical Bending of Functionally Graded Piezoelectric Material Plates北大核心CSCD

功能梯度压电材料（FGPM）同时兼具功能梯度材料和压电材料特性,可为多功能或智能化轻质结构设计提供支撑,在诸多领域有着广泛的应用前景.将Mian和Spencer功能梯度板理论由功能梯度弹性材料推广到功能梯度压电材料,解析研究了FGPM板的柱面弯曲问题,其中,材料弹性常数、压电和介电参数沿板厚方向可以任意连续变化.最终,给出了FGPM板受横向均布荷载作用下柱面弯曲问题的弹性力学解.通过算例分析,重点讨论了压电效应对FGPM板静力响应的影响.     

The solution of unsteady coupled electromechanical problems for multilayered media

A rigorous formulation of the unsteady coupled electromechanical problem of the interaction of a massive electrode with a multilayered piezoelectric medium is given, and a general formulation of a method of solving it is proposed. As an example the problem for a massive strip electrode, which interacts with a single-layer, double-layer and triple-layer piezoelectric medium with unsteady loading is considered. The effect of the electroelastic properties of different materials (classes 6 mm of the hexagonal system) on the displacement of the electrode and the potential is pointed out.     

Local solutions to a model of piezoelectric materials

A local existence theorem is proved for a non‐linear coupled system modelling the electromechanical motion of a one‐dimensional piezoelectric body with domain switching. The system is composed by a heat equation describing the behaviour of the number of electric dipoles and by a wave equation governing the dynamic of the electric displacement. The main coupling in the system appears in the time‐dependent velocity of the waves depending on the number of electric dipoles. The proof of the result relies on a time decay estimate satisfied by the number of electric dipoles and an uniform estimate of the solution of the regularized wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

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)     

Analysis of interface cracks in layered piezoelectric composites by a SGBEM

Piezoelectric materials offer many possibilities in advanced engineering structures due to their inherent coupling effects between mechanical and electrical fields and are widely applied in smart devices and structures like transducers, actuators and sensors [2]. An important application of piezoelectric materials is related to layered or laminated composites because they can be optimized to satisfy the high-performance requirements according to different in-service conditions. Beside cracks inside homogeneous domains, one of the most dominant failure mechanisms in layered or laminated composites is the interface failure. Interface cracks and interface debonding may be induced by the mismatch of the mechanical, electrical and thermal properties of the material constituents during the manufacturing process and the in-service loading conditions. This paper presents a hypersingular symmetric Galerkin boundary element method (SGBEM) for crack analysis in two-dimensional (2D), layered and linear piezoelectric solids. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Coupled FE-BE Analysis of Smart Lightweight Structures for Active Noise and Vibration Control

Over the past years much research and development has been done in the area of active control in order to improve the acoustical and vibrational properties of thin–walled lightweight structures. An efficient technique for actively reducing the structural vibration and sound radiation is the application of smart structures. In smart structures piezoelectric materials are often used as actuators and sensors. The design of smart structures requires fast and reliable simulation tools. Therefore, the purpose of this paper is to present a coupled finite element–boundary element formulation, which enables the modeling of piezoelectric smart lightweight structures. The paper describes the theoretical background of the coupled approach in which the finite element method (FEM) is applied for the modeling of the passive vibrating shell structure as well as the surface attached piezoelectric actuators and sensors. The boundary element method (BEM) is used to characterize the corresponding sound field. In order to derive a coupled FE–BE formulation additional coupling conditions are introduced at the fluid–structure interface. Since the resulting overall model contains a large number of degrees of freedom, the mode superposition method is employed to reduce the size of the FE submodel. To validate the accuracy of the proposed approach, numerical simulations are carried out in the frequency domain and the results are compared with analytical reference solutions. (© 2008 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)