A hybrid phenomenological model for ferroelectroelastic ceramics. Part I: Single phased materials

In this part I of a two part series, a rate-independent hybrid phenomenological constitutive model applicable for single phased polycrystalline ferroelectroelastic ceramics is presented. The term “hybrid” refers to the fact that features from macroscopic phenomenological models and micro-electromechanical phenomenological models are combined. In particular, functional forms for a switching function and the Helmholtz free energy are assumed as in many macroscopic phenomenological models; and the volume fractions of domain variants are used to describe the internal material state, which is a key feature of micro-electromechanical phenomenological models. The approach described in this paper is an attempt to combine the advantages of macroscopic and micro-electromechanical material models. Its potential is demonstrated by comparison with experimental data for barium titanate. Finally, it is shown that the model for single phased materials cannot reproduce the material behavior of morphotropic PZT ceramics based on a realistic choice for the material parameters. This serves as a motivation for part II of the series, which deals with the modeling of morphotropic PZT ceramics taking into account the micro-structural specifics of these materials.     

Two-scale micromechanics-based probabilistic modeling of domain switching in ferroelectric ceramics

A two-scale micromechanics model is developed in this paper to analyze domain switching in ferroelectric ceramics, using a probabilistic domain switching criterion based on energetic analysis. The microstructure of ferroelectric ceramics at two distinct length scales, domains and grains, has been carefully analyzed. The interaction at domain level is accounted for by energy minimization theory, while the fluctuation at grain level is analyzed using ellipsoidal two-point correlation function. The model has been implemented by Monte Carlo method, and applied to simulate the electric poling and mechanical depoling of Pb(Zr_xTi_1-x)O₃ (PZT) ceramics across morphotropic phase boundary (MPB). The drastically different switching characteristics of PZT ceramics across MPB has been captured, and good agreement with experiments has been observed. The effects of the transformation strains and spontaneous polarizations are highlighted, confirming the proposition of Li et al. [2005. Domain switching in polycrystalline ferroelectric ceramics. Nature Materials 4, 776–781] that the strain compatibility plays a dominant role in domain switching in ferroelectric ceramics.     

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.     

An optical method to assess electromechanical coupling in ferroelectric ceramics

A new technique is described, which allows the assessment of elastic and inelastic regions around a macroscopic defect in ferroelectric-ferroelastic ceramics. The accuracy and robustness of the method are demonstrated on a PZT plate with a centered hole subjected to uni-axial compressive stresses. From the electrical potential distribution on the sample surface, the mechanical response of the material is obtained at different load levels.     

On twinning and domain switching in ferroelectric Pb(Zr1−xTix)O3—part I: twins and domain walls

We study the electromechanical behavior of lead zirconate titanate ferroelectric ceramics (PZT), by means of a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by spontaneous polarization and strain. Spontaneous polarization and strain organize into a domain structure which minimizes electrostatic and elastic energies and which can be modified by the application of electromechanical loads. Such process, which is called “domain switching”, is associated with electrical and mechanical hysteresis and can be studied as a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics. In this paper, which is the first of two, we deal with the electromechanical model and related constitutive assumptions, as well as with the analysis of domain structure in PZT. In particular, following the discover of a new monoclinic phase in PZT carried by Noheda and co-workers, we analyze twinning between spontaneous strain at the various phase boundaries and show that both non-generic, non-conventional twins and finely-twinned laminates are possible, and also that the presence of a monoclinic phase may explain PZT exceptional properties.     

Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics

In this paper, a general form for multi-axial constitutive laws for ferroelectric ceramics is constructed. The foundation of the theory is an assumed form for the Helmholtz free energy of the material. Switching surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed. The resulting tangent moduli relating increments of stress and electric field to increments of strain and electric displacement are symmetric since changes in the linear elastic, dielectric and piezoelectric properties of the material are included in the switching surface. Finally, parameters of the model are determined for two uncoupled cases, namely non-remanent straining ferroelectrics and purely ferroelastic switching, and then for the fully coupled ferroelectric case.     

Development, modeling and deflection analysis of hybrid micro actuator with integrated thermal and piezoelectric actuation

Micro actuators are irreplaceable part of motion control in minimized systems. The current study presents an analytical model for a new Hybrid Thermo Piezoelectric micro actuator based on the combination of piezoelectric and thermal actuation mechanisms. The micro actuator structure is a double PZT cantilever beam consisting of two arms with different lengths. The presented micro actuator uses the structure of electrothermal micro actuator in which polysilicon material is replaced by PZT. Also the voltage and poling directions are considered in the lengthwise of PZT beams. As a result, the piezoelectric actuation mechanism is based on d ₃₃ strain coefficient. The tip deflection of micro actuator is obtained using Timoshenko beam theory. Analytical results are compared with FEM results along with other reported results in the literature. The effects of geometrical parameters and PZT material constants on actuator tip deflection are studied to provide an efficient optimization of HTP micro actuator.     

柱形爆电换能器的理论研究

本文根据冲击波加载铁电陶瓷的换能原理,建立了斜冲击波加载PZT柱形组件的电响应珲论模型。所考虑的换能装置特点是:(1)柱形组件由n块PZT空心圆盘并联堆叠而成:(2)PZT样品在冲击波压缩区具有恒定电导率r:(3)外电路由串联的电感L和电阻R以及并联电容C_p组成。数值计算结果表明,这种柱形爆电换能器用来提供大电流,大能量(如2000A、100J)是适宜的。     

多孔极化PZT95/5铁电陶瓷单轴压缩力学响应与放电特性

采用添加造孔剂的方法制备了四种不同孔隙率PZT95/5铁电陶瓷,对其进行电场极化,随后开展了准静态单轴压缩实验,讨论了畴变、相变以及孔隙率对极化PZT95/5铁电陶瓷的力学响应与放电特性的影响. 研究结果表明：(1)多孔极化PZT95/5铁电陶瓷非线性力学响应行为主要归因于畴变和相变的共同作用,与内部孔洞变形和坍塌基本无关;(2) 在准静态单轴压缩下极化PZT95/5铁电陶瓷的去极化机制是畴变和相变的共同作用;(3) 孔隙率对极化PZT95/5铁电陶瓷的弹性模量、压缩强度有明显的影响,而对断裂应变的影响较小;(4)极化PZT95/5铁电陶瓷畴变和相变开始的临界应力都随着孔隙率的增大而线性衰减,但相变开始的临界体积应变却不依赖孔隙率;(5)极化PZT95/5铁电陶瓷电荷饱和释放量随着孔隙率呈线性减小,但孔隙率对电荷释放速率基本没有影响。     

Elasticity and strength of partially sintered ceramics

A discrete element model for the elastic and fracture behavior of partially sintered ceramics is presented. It accounts for the granular character of the material when a large amount of porosity (typically >0.2-0.4) is left after sintering. The model uses elastic force-displacement laws to represent the bond formed between particles during sintering. Bond fracture in tension and shearing is accounted for in the model. Realistic numerical microstructures are generated using a sintering model on random particle packings. In particular, packings with fugitive pore formers are used to create partially sintered microstructures with large pores. The effective elastic response and the strength of these microstructures are calculated in tension and compression. The link between important microstructural features such as bond size or coordination number and macroscopic behavior is investigated. In particular, it is shown that porosity alone is not sufficient to account for the mechanical properties of a partially sintered material.     

高应变率下多孔未极化PZT95/5铁电陶瓷的非线性力学行为

采用添加造孔剂的方法制备了4种不同孔隙率的未极化PZT95/5铁电陶瓷。采用基于超高速相机与数字图像相关性方法的试样全场应变测量技术以及分离式霍普金森压杆（SHPB）技术,对多孔未极化PZT95/5铁电陶瓷进行高应变率单轴压缩实验研究。全场应变测量结果显示:轴向应变仅在试样中部分布较均匀,将该区域的平均应变作为应力-应变关系中的试样应变测量值较为合理,而由SHPB原理计算的试样应变值明显偏大,需要摒弃或修正传统的SHPB数据处理方法。通过波形整形技术实现了恒应变率加载,弱化了径向惯性效应的影响,揭示出多孔未极化PZT95/5铁电陶瓷的压缩强度具有显著的应变率效应。通过分析试样轴向应变和径向应变随着加载应力的变化,阐明多孔未极化PZT95/5铁电陶瓷的非线性变形行为的物理机制是畴变和相变共同作用,并发现畴变临界应力和相变临界应力都随着应变率升高而增大。保持加载应变率不变,讨论了孔隙率对多孔未极化PZT95/5铁电陶瓷动态力学行为的影响,发现随着孔隙率的升高,动态压缩强度呈非线性衰减,而畴变临界应力和相变临界应力则基本呈线性衰减。     

Stiffness and strength of phase transformation ceramics containing misoriented microcracks

Microcracks exert a large influence on the behavior of phase transformation ceramics. In this paper, the cause of microcracking is interpreted and the interaction between transformation particles and microcracks is described to estimate the stiffness and strength through the modified equivalent inclusion method by the authors. Three point bend experiments on Al₂O₃/ZrO₂ ceramics are performed and a comparison between the theoretical and measured results confirms the rationality of the analytical model proposed in this paper and the important role of microcracks.     

Phenomenological model for the macroscopical material behavior of ferroelectric ceramics

A thermodynamically consistent phenomenological model for the simulation of the macroscopic behavior of ferroelectric polycrystalline ceramics is presented. It is based on the choice of microscopically motivated internal state variables, which describe the texture and the polarization state of the polycrystal. Saturation states are defined for the internal state variables. The linear material behavior is modelled by a transversely isotropic piezoelectric constitutive law, where the anisotropy is history dependent. For non-linear irreversible processes, a switching function and associated evolution rules are applied, satisfying the principle of maximum ferroelectric dissipation. Saturation is modelled by the use of energy-barrier functions in the electric enthalpy density function. Numerical examples demonstrate the capability of the proposed model, to predict the typical experimental results.     

Effect of electric displacement saturation on the hysteretic behavior of ferroelectric ceramics and the initiation and propagation of cracks in piezoelectric ceramics

This paper presents a computational investigation of a proposed simplified account for electric displacement saturation on the hysteretic behavior of initially unpoled ferroelectric ceramics as well as on the initiation and propagation of cracks in poled ferroelectric ceramics within the linear regime of piezoelectricity. For the latter case, experimental observations suggest an odd dependency of the onset of crack initiation in these brittle materials on the orientation of the applied electric field with respect to their poling direction which contradicts theoretical results which propose an even dependency of the energy release rate on the applied electric field within the framework of anisotropic linear piezoelectricity. Electric non-linearities arising at regions of inhomogeneities such as inclusions or at the crack tip are proposed in the literature to avoid this discrepancy. Electric displacement saturation is one such non-linear effect which is investigated in this work. A simplified account of this effect is proposed based on an exponential saturation model of the identified material parameters which can be related to this non-linearity. Its advantage over the superposition of a complex function onto the singular solution of a crack within the framework of linear piezoelectricity lies in the straightforward extension of the proposed approach to problems where no analytical solutions exist. This is outlined based on its incorporation into a rate-dependent ferroelectric model accounting for polarization switching as well as based on its incorporation into a finite element framework capable of simulating the initiation and propagation of cracks in piezoelectric ceramics through strong discontinuities in the displacement field and the electric potential. It is shown that besides the determination of the crack initiation onset also the crack propagation direction is influenced by the appearance of saturation zones arising at the crack tip normal to the polarization direction. The numerically obtained crack paths are found to be close to the experimentally reported results.     

A one-dimensional continuum model for ferroelectric ceramics

In this article, we introduce a one-dimensional continuum model for ferroelectric ceramics within a thermodynamical framework. The model consists of a free energy potential, a switching criterion, and a kinetic relation. The free energy potential is given as a function of polarization, strain, and two internal variables – remanent polarization and remanent strain. A polarization switching is described by evolutions of the two internal variables and evolution laws called kinetics are proposed based on the second law of thermodynamics. The predictions of the model are compared with experimental observations. It is suggested to model unpoled domains in the fully poled state for improved model responses.     

Damage,plasticity and failure of ceramics and cementitious composites subjected to multi-axial state of stress

Experimental data on mechanical behavior of ceramics and cementitious composites subjected to triaxial state of stress and verification of the theoretical model capable to describe deformability and fracture of brittle rock-like materials are presented in the paper. To check the validity of the theoretical model the stress–strain curves and stresses at material fracture determined experimentally for brick and mortar were compared with the theoretical predictions. The limit surface at material fracture obtained experimentally from triaxial tests was used in numerical analysis of masonry specimens subjected to compressive loading. These numerical results obtained by employing the Finite Element Method software package Mafem3D were compared with experimental data available in the literature. Fairly good agreement of numerical predictions with experimental results for masonry specimens was observed.     

Electric-field-induced fatigue crack growth in ferroelectric ceramics

Electric-field-induced fatigue crack growth in pre-cracked PZT ferroelectric ceramics is experimentally investigated in this work. It is found that the crack open and close under an alternating electric field is a major mechanism of crack propagation. The experimental results also show that the frequency, waveform, as well as the amplitude ratio, of the electric loading, play important roles in electric-field-induced fatigue cracking. Empirical formulations of fatigue crack propagation rates are obtained based on the experimental results. It is revealed that the crack grows at a nearly constant rate when the loading frequency is below 100 Hz. However, with the increase of the loading frequency over 125 Hz, the crack propagation rate diminishes rapidly.     

Time dependence of effective properties of polycrystalline ferroelectric ceramics

In this paper, based on Merz[7] experimental results and classical nucleation theory, a micromechanics statistical model is proposed to describe the relation between the special microstructure-level evolution phenomena-domain switching and macro-response. The polycrystalline ferroelectric ceramics treated as a composition of switched domain and unswitched domain, the approaches of Eshelby's equivalent inclusion and Mori-Tanaka's mean field theory are used to analyze and predict its effective electroelastic properties. The model can incorporate the effects of time dependence of domain switching and shape of individual crystalline. To the BaTiO₃ polycrystalline ceramics, the analytical results are in good agreement with the experimental results.     

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.     

主动约束层阻尼梁的数值模型

为对主动约束层阻尼结构建立精确完善的数学模型，采用有限元建模，并考虑到压电材料的机电耦合效应和粘弹性材料的本构关系随温度、频率的变化而变化的特点，将有限元方法与粘弹性材料的GHA模型相结合，从而避免因粘弹性材料导致的非线性微分方程，能直接求解模态频率、模态阻尼及结构响应。为进一步设计控制器，先在物理空间进行动力缩聚，将系统降至适当的维数，然后在状态空间用鲁棒防阶的方法进一步降阶。这样既能大大降低系统维数，又能保证降阶后系统稳定、可控、可观。这对于重量轻、柔度大、低频密集的大型空间柔性结构尤其重要。