Effect of geometry upon the performance of a thin film ferroelectric capacitor

The finite element method is used to investigate the performance of a ferroelectric random access memory as a function of its geometry. Performance is characterised by the charge versus electric field relation, and the sensitivity of performance to geometry is explored. The primary geometric variables are the dimensions of a prismatic two-dimensional (2D) island of ferroelectric material, and the edge inclination angle caused by the etching process along the sides of the island. The performance of the two-dimensional ferroelectric device is compared to those of an unsupported ferroelectric thin film and of a ferroelectric film bonded to a substrate.     

Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading

Complex, non-linear, irreversible, hysteretic behavior of polycrystalline ferroelectric materials under a combined electro-mechanical loading is a result of domain wall motion, causing simultaneous expansion and contraction of unlike domains, grain sub-divisions that have distinct spontaneous polarization and strain. In this paper, a 3-dimensional finite element method is used to simulate such a polycrystalline ferroelectric under electrical and mechanical loading. A constitutive law due to Huber et al. [1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47, 1663-1697] for switching by domain wall motion in multidomain ferroelectric single crystals is employed in our model to represent each grain, and the finite element method is used to solve the governing conditions of mechanical equilibrium and Gauss's law. The results provide the average behavior for the polycrystalline ceramic. We compare the outcomes predicted by this model with the available experimental data for various electromechanical loading conditions. The qualitative features of ferroelectric switching are predicted well, including hysteresis and butterfly loops, the effect on them of mechanical compression, and the response of the polycrystal to non-proportional electrical loading.     

A finite element model of ferroelastic polycrystals

A finite element model of switching in polycrystalline ferroelastic ceramics is developed. It is assumed that a crystallite switches if the reduction in mechanically driven potential energy of the system exceeds a critical value per unit volume of switching material. Stress induced (i.e. ferroelastic) switching is a change of permanent strain in characteristic crystallographic directions. Martensitic twinning is one example, but the strain response of ferroelectric materials has the same characteristics. The model is suitable for representing ferroelastic systems such as shape memory alloys and as a preliminary model for ferroelectric/ferroelastic materials such as perovskite piezoelectrics. In the simulations, each crystallite is represented by a finite element and the crystallographic principal direction for each crystallite is assigned randomly. Different critical values for the energy barrier to switching are selected to simulate stress vs strain hysteresis loops of a ceramic lead lanthanum zirconate titanate (PLZT) at room temperature. The measured stress versus strain curves of polycrystalline ceramics designated PZT-A and PZT-B are also reproduced by the model.     

A ferroelectric and ferroelastic 3D hexahedral curvilinear finite element

An isoparametric 3D electromechanical hexahedral finite element integrating a 3D phenomenological ferroelectric and ferroelastic constitutive law for domain switching effects is proposed. The model presents two internal variables which are the ferroelectric polarization (related to the electric field) and the ferroelastic strain (related to the mechanical stress). An implicit integration technique of the constitutive equations based on the return-mapping algorithm is developed. The mechanical strain tensor and the electric field vector are expressed in a curvilinear coordinate system in order to handle the transverse isotropy behavior of ferroelectric ceramics. The hexahedral finite element is implemented into the commercial finite element code Abaqus® via the subroutine user element. Some linear (piezoelectric) and non linear (ferroelectric and ferroelastic) benchmarks are considered as validation tests.     

A phenomenological constitutive model for ferroelastic and ferroelectric hysteresis effects in ferroelectric ceramics

This paper is concerned with a macroscopic constitutive law for domain switching effects, which occur in ferroelectric ceramics. The three-dimensional model is thermodynamically consistent and is determined by two scalar valued functions: the Helmholtz free energy and a switching surface. In a kinematic hardening process the movement of the center of the switching surface is controlled by internal variables. In common usage, the remanent polarization and the irreversible strain are employed as internal variables. The novel aspect of the present work is to introduce an irreversible electric field, which serves instead of the remanent polarization as internal variable. The irreversible electric field has only theoretical meaning, but it makes the formulation very suitable for a finite element implementation, where displacements and the electric potential are the nodal degrees of freedom. The paper presents an appropriate implementation into a hexahedral finite brick element. The uni-axial constitutive model successfully reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for ferroelectric ceramics. Furthermore it accounts for the mechanical depolarization effect, which occurs if the polarized ferroelectric ceramic is subjected to a compression stress.     

Micromechanical modelling of remanent properties of morphotropic PZT

The present paper investigates the capability of micromechanical material models to predict the ferroelectric behaviour of morphotropic PZT ceramics in a rate-independent approximation based on realistic microscopic material parameters. Starting point is a three-dimensional tetragonal model, which builds on the model of Pathak and McMeeking [2008. Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading. Journal of the Mechanics and Physics of Solids 56, 663-683]. Volume fractions of the crystallographic variants represent the domain structure inside the grains. Interactions between the grains are taken into account by means of a representative volume element of the grain compound. A simplified set of realistic microscopic material parameters of the lattice in terms of Young's modulus, Poisson's ratio, dielectric constant, and spontaneous strain and polarisation is derived from experimental data and theoretical results given in the literature. The simulation of the macroscopic remanent polarisation and strain response due to two load cases shows explicitly that the tetragonal model is not capable to reproduce the behaviour of morphotropic PZT. Therefore, the model is extended by the rhombohedral phase, allowing a mixture of both phases with varying quantities inside the grains. A comparison of our results with experimental data shows a remarkably good agreement, revealing the capability of the extended model.     

Multi-grain analysis versus self-consistent estimates of ferroelectric polycrystals

Ferroelectrics are polycrystalline materials consisting of intragranular regions with different polarization directions, called domains. The domains can be switched into different states by the application of an electric field or mechanical stress. We study the influence of grain-to-grain interactions on the overall and local switching behavior. The behavior inside each grain is represented by the micromechanics model of Huber et al. [1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47 (8), 1663-1697]. The predictions of a self-consistent model of the polycrystal response are compared with those of a multi-grain model in which grains are represented individually. In one flavor of the multi-grain model, each grain is represented by a single finite element, while in the other the fields inside each grain are captured in more detail through a fine discretization. Different electrical and mechanical loading situations are investigated. It is found that the overall response is only mildly dependent on the accuracy with which grain-to-grain interactions are captured, while the distribution of grain-average stresses is quite sensitive to the resolution of the intragranular fields.     

Finite element simulation of a polycrystalline ferroelectric based on a multidomain single crystal switching model

In this paper, we compute the constitutive behavior of a ferroelectric ceramic by a plane strain finite element model, where each element represents a single grain in the polycrystal. The properties of a grain are described by the microscopic model for switching in multidomain single crystals of ferroelectric materials presented by Huber et al. [J. Mech. Phys. Solids 47 (1999) 1663]. The poling behavior of the polycrystal is obtained by employing the finite element formulation for electromechanical boundary value problems developed by Landis [Int. J. Numer. Meth. Eng. 55 (2002) 613]. In particular, we address the influence of the single grain properties and the interaction between grains, respectively.     

A finite element model for rate-dependent behavior of ferroelectric ceramics

In this article, materials within a crystallite are modeled by continuum particles consisting of various types of ferroelectric variants which are characterized by their mass fractions. The constitutive behavior of each type of variant is characterized by a proposed Helmholtz free energy potential. Polarization switching is modeled by continuous changes of mass fractions which are governed by a onset criterion and a kinetic relation. A finite element algorithm is developed using the virtual work principle. The simulated results on the rate dependence in the polarization and strain responses to applied alternating electric field of different frequencies are in qualitative consistence with experimental observations. The rate-dependent behavior is explained in terms of changes of mass fractions of the variants that polarization switching involves, in response to the loading programs of different loading rates.     

Numerical study of the plastic behaviour in tension of welds in high strength steels

The influence of the mismatch between material properties and constraint on the plastic deformation behaviour of the heat affected zone of welds in high strength steels is investigated in this study, using finite element simulations. An elastoplastic implicit three-dimensional finite element code (EPIM3D) was used in the analysis. The paper presents the mechanical model of the code and the methodology used for the numerical simulation of the tensile test of welded joints. Numerical results of the tensile test of welded samples with different hypothetical widths for the Heat Affected Zone and various material mismatch levels are shown. The analysis concerns the overall strength and ductility of the joint and in relation to the plastic behaviour of the heat affected zone. The influence of the yield stress, tensile strength and constraint on the stress and plastic strain distribution in the soft heat affected zone is also discussed.     

Numerical simulation of domain switching in multilayer ferroelectric actuators

Micromechanical finite element methods are developed based on a nonlinear constitutive model of ferroelectric polycrystals. Electromechanical behaviors ahead of an internal electrode tip are numerically simulated in multilayer ferroelectric actuators. Around the electrode edge, the nonuniform electric field generates a concentration of stress due to the incompatible strain as well as spontaneous strain. The preferred domain switching enhances the concentration of residual stress and may cause the actuators to crack. An electrically permeable crack emanating from an internal electrode is analyzed. A large scale domain switching zone is found in the vicinity of crack tips. The larger the actuating strain and electric field are, the larger the switching zone will be. The size of switching zone even reaches the scale of crack length with increasing electromechanical loading.     

微结构片外弯曲测试装置及试样设计

硅是微电子机械系统（简称微机电系统）中最常见的功能结构材料,可靠性是制约硅微构件小尺度加工和大规模制造的瓶颈问题．为研究硅微构件的力学特性,本文开发了一套以压电驱动、微力测量、位移检测为核心组件的片外测试系统．设计了一种将四个弯曲测试梁集于一体的微结构,借助有限元方法确定其尺寸,并用理论方法验证有限元分析的合理性．本文着重确保了四个关键设计目标：一、每根测试梁最大应力应位于其与外框架结合处;二、未断裂测试梁的最大应力受其他梁的断裂的影响应足够小;三、各个测试梁的最大应力的差别应足够小;四、支撑梁的最大应力应明显小于测试梁．最后测试了试样的弯曲强度,实验加载曲线和有限元分析基本吻合,表明测试装置和试样设计是合理的,为后续的硅微构件可靠性测试奠定了基础．     

Analysis of the electromechanical behavior of ferroelectric ceramics based on a nonlinear finite element model

A nonlinear finite element (FE) model based on domain switching was proposed to study the electromechanical behavior of ferroelectric ceramics. The incremental FE formulation was improved to avoid any calculation instability. The problems of mesh sensitivity and convergence, and the efficiency of the proposed nonlinear FE technique have been assessed to illustrate the versatility and potential accuracy of the said technique. The nonlinear electromechanical behavior, such as the hysteresis loops and butterfly curves, of ferroelectric ceramics subjected to both a uniform electric field and a point electric potential has been studied numerically. The results obtained are in good agreement with those of the corresponding theoretical and experimental analyses. Furthermore, the electromechanical coupling fields near (a) the boundary of a circular hole, (b) the boundary of an elliptic hole and (c) the tip of a crack, have been analyzed using the proposed nonlinear finite element method (FEM). The proposed nonlinear electromechanically coupled FEM is useful for the analysis of domain switching, deformation and fracture of ferroelectric ceramics.The project supported by the National Natural Science Foundation of China (10025209, 10132010 and 90208002), the Research Grants of the Council of the Hong Kong Special Administrative Region, China (HKU7086/02E) and the Key Grant Project of the Chinese Ministry of Education (0306)     

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.     

Thermodynamics of Shape Memory Alloy Wire: Modeling, Experiments, and Application

A thermomechanical model for a shape memory alloy (SMA) wire under uniaxial loading is implemented in a finite element framework, and simulation results are compared with mechanical and infrared experimental data. The constitutive model is a one–dimensional strain-gradient continuum model of an SMA wire element, including two internal field variables, possible unstable mechanical behavior, and the relevant thermomechanical couplings resulting from latent heat effects. The model is calibrated to recent and new experiments of typical commercially available polycrystalline NiTi wire. The shape memory effect and pseudoelastic behaviors are demonstrated numerically as a function of applied displacement rate and environmental parameters, and the results compare favorably to experimental data. The model is then used to simulate a simple SMA actuator device, and its performance is assessed for different thermal boundary conditions.     

Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory

A finite element formulation of the piezoelectric vibrations of quartz resonators based on Mindlin plate theory is derived. The higher-order plate theory is employed for the development of a collection of successively higher-order plate elements which can be effective for a broad frequency range including the fundamental and overtone modes of thickness-shear vibrations. The presence of electrodes is also considered for their mechanical effects.The mechanical displacements and electric potential are combined into a generalized displacement field, and the subsequent derivations are carried out with all the generalized equations. Through the standard finite element procedure, the vibration frequency, the vibration mode shapes and the electric potential distribution are obtained. The frequency spectra are compared with some well-known experimental results with good agreement.Our previous experience with finite element analysis of high-frequency quartz plate vibrations leads us to believe that memory and computing time will always remain as key issues despite the advances in computers. Hence, the use of sparse matrix techniques, efficient eigenvalue solvers, and other reduction procedures are explored.     

The stiffness and strength of the gyroid lattice

Recently, a nanoscale lattice material, based upon the gyroid topology has been self-assembled by phase separation techniques (Scherer et al., 2012) and prototyped in thin film applications. The mechanical properties of the gyroid are reported here. It is a cubic lattice, with a connectivity of three struts per joint, and is bending-dominated in its elasto-plastic response to all loading states except for hydrostatic: under a hydrostatic stress it exhibits stretching-dominated behaviour. The three independent elastic constants of the lattice are determined through a unit cell analysis using the finite element method; it is found that the elastic and shear modulus scale quadratically with the relative density of the lattice, whereas the bulk modulus scales linearly. The plastic collapse response of a rigid, ideally plastic gyroid lattice is explored using the upper bound method, and is validated by finite element calculations for an elastic-ideally plastic lattice. The effect of geometrical imperfections, in the form of random perturbations to the joint positions, is investigated for both stiffness and strength. It is demonstrated that the hydrostatic modulus and strength are imperfection sensitive, in contrast to the deviatoric response. The macroscopic yield surface of the imperfect lattice is adequately described by a modified version of Hill’s anisotropic yield criterion. The article ends with a case study on the stress induced within a gyroid thin film, when the film and its substrate are subjected to a thermal expansion mismatch.     

Simulation of butterfly loops in ferroelectric materials

Ferroelectric materials exhibit complex behaviour upon electric and mechanical loading. Their change of polarization and length is accompanied by hysteresis. The so-called butterfly loop characterizing the strain due to an applied electric field is striking. As these materials are used in technical applications as sensors, actuators, and non-volatile memory units, the ability to simulate these hysteretic phenomena is important. Here, a one-dimensional model is proposed that is capable of describing the qualitative behaviour of widely used perovskite-type ferroelectric materials.Received: 3 July 2003, Accepted: 10 July 2003     

铁电薄膜在红外辐射作用下的动力响应

本文研究了红外探测器中红外焦平面列阵的象元———铁电薄膜微桥在红外辐射作用下的输出信号。用层合板壳作为微桥结构的力学模型,中间一层为压电材料,上下两层为金属材料电极。采用了力-电-热耦合的控制方程和变分原理,考虑了铁电薄膜的惯性力,推导出了基于Mindlin假设的压电材料层合板有限元公式。以红外探测器在夜间从飞机上探测地面的坦克为例,用有限单元法模拟了铁电薄膜微桥在红外辐射作用下的力、热、电输出信号,并对结果作了分析比较。     

铁电陶瓷宏观单轴力电行为的双面模型

铁电陶瓷以其优越的力电耦合性作为新型的智能材料使用. 提出基于弹塑性双面理论的宏观铁电本构模型. 根据铁电陶瓷内部电畴在外电场和机械场作用下的微观运动，在宏观上除引入材料的畴变面外，还首次引入饱和面，并考虑以畴变面与饱和面之间的广义距离来表征铁电陶瓷的非线性行为. 数值计算结果与实验数据的比较表明所提出的初步理论可适当地反映力电加载下铁电陶瓷的宏观非线性行为.