Relaxation model of piezoelectric materials

Polarization switching inside grains is time dependent. When external applied loading is not quasi-static, macroscopic properties of piezoelectric materials changes with the rate of loading. In this paper, a 2-D micromechanical model is proposed in order to simulate the rate dependent properties of certain perovskite type tetragonal piezoelectric materials based on linear constitutive, nonlinear domain switching, intergranular effects and kinetics models. The material is electrically loaded with an alternating voltage of various frequencies. For the onset of domain switching, energy equation is implemented. Propagation of the domain wall during domain switching in grains is modeled by means of exponential kinetics relation after domain nucleation. Mechanical strain butterfly loops under different frequencies (0.01Hz–1Hz) are simulated. The model gives important insights into the rate dependency of the piezoelectric materials that have been observed in some experiments reported in the literature. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

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)     

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)     

A laminate-based framework for switching and microstructure evolution in polycrystalline ferroelectrics

This contribution deals with the numerical modelling of polycrystalline ferroelectric materials considering a sequential laminate-based approach established for tetragonal single-crystal ferroelectrics. The particular model [1] is considered and extended to predict the material behaviour of poly-crystal tetragonal ferroelectric ceramics. The derived laminate-based model is implemented in a finite element environment to simulate the time-dependent domain evolution and switching response of a bulk polycrystalline ferroelectric ceramic. © 2016 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)     

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)     

Model selection of a switching mechanism for financial time series

The threshold autoregressive model with generalized autoregressive conditionally heteroskedastic (GARCH) specification is a popular nonlinear model that captures the well‐known asymmetric phenomena in financial market data. The switching mechanisms of hysteretic autoregressive GARCH models are different from threshold autoregressive model with GARCH as regime switching may be delayed when the hysteresis variable lies in a hysteresis zone. This paper conducts a Bayesian model comparison among competing models by designing an adaptive Markov chain Monte Carlo sampling scheme. We illustrate the performance of three kinds of criteria by comparing models with fat‐tailed and/or skewed errors: deviance information criteria, Bayesian predictive information, and an asymptotic version of Bayesian predictive information. A simulation study highlights the properties of the three Bayesian criteria and the accuracy as well as their favorable performance as model selection tools. We demonstrate the proposed method in an empirical study of 12 international stock markets, providing evidence to strongly support for both models with skew fat‐tailed innovations. Copyright © 2016 John Wiley & Sons, Ltd.     

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)     

The radial basis reproducing kernel particle method for geometrically nonlinear problem of functionally graded materials

In this paper, the radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is proposed for solving geometrically nonlinear problem of functionally graded materials (FGM). Compared with the RKPM, the advantages of the proposed method are that it can eliminate the negative effect of different kernel functions on the computational accuracy, and has higher computational accuracy and stability. Using the Total Lagrange (T.L.) formulation and the weak form of Galerkin integration, the corresponding formulae for geometrically nonlinear problem of FGM are derived. The penalty factor, shaped parameter of the RBF, the control parameter of influence domain radius, loading step number and node distribution are discussed. Furthermore, the effects of different gradient functions and exponents on displacement and stress are analyzed. Newton-Raphson (N-R) iterative method is utilized for numerical solution. The proposed method is correct and effective for solving geometrically nonlinear problem of FGM, which can be demonstrated by several numerical examples.     

Comparative micromechanical analysis of nonelastic behavior of unidirectional plastics with tetragonal and hexagonal structures

A mathematical model for determining the effective elastic properties and describing the processes of inelastic deformation and damage accumulation of unidirectional fiber-reinforced composites with tetragonal and hexagonal structures is developed. A comparative analysis of the effective elastic moduli of glass, boron, organic, and carbon unidirectional plastics shows that, if the fiber volume fraction does not exceed 0.5, the effective elastic properties calculated by the models presented give closely related results. The calculation results for nonlinear fields of deformation and failure are presented and the limiting strength surfaces of fibrous glass plastics with hexagonal and tetragonal structures are obtained for different transverse loading paths. It is found that the structure of a composite affects significantly its strength properties.Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Perm' State Technical University, Perm', Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 455–464, July–August, 2000.     

Predicting the transverse strength of unidirectional composites under combination loading

This article presents a mathematical model for predicting the transverse strength of unidirectional fiber composites subjected to combination transverse loading under different conditions. The behavior of the matrix is described by nonlinear physical equations consistent with the strain theory of plasticity for the active loading section. The fibers are assumed to be isotropic and elastic. The boundary-value problem of micromechanics that is formulated includes strength criteria for the matrix and fibers that mark the beginning of their possible failure. The modeling of the fracture process is taken farther through the use of a scheme that reduces the stiffness of the matrix and fibers in the failed regions in relation to the sign of the first invariant of the stress tensor. The method of local approximation is used together with the finite-element method to calculate the stress and strain fields in unidirectional composites with cylindrical fibers in a tetragonal layup. The model is used to study the behavior of an epoxy-based organic-fiber-reinforced plastic subjected to transverse loading in different simple paths — including simultaneous compressive and tensile loads, as well as transverse shear.Paper to be presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 473–481, July–August, 1995.     

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。     

An estimation of reliability of unidirectional composites by catastrophe theory

In the present work, interest is centered on the theory of fracture and practical approaches to reliability estimation of unidirectional composite materials which are based on it. Fracture mechanics is considered not as a theory of macrocracks but as mechanics of fracture mechanisms of the composites with allowance made for probability estimation. A model of composite material with fibers eliminated from the carrying scheme and parameters specific to the stress-strain-damage state (SSDS) was considered. According to the model, during loading, self-accelerated energy rise is accounted for by structural damages growth — the catastrophe takes place. The model allows us to calculate the critical stress. The composites redistribute forces from overloading zones to neighboring ones by microstructural deformations and damages. It is the effect of reservation of carrying ability. Due to this effect, low-value probability of fracture is sufficiently less for the composite than for homogeneous materials. The approach allows us to evaluate the reliability function for both static loading and fatigue.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Computing Center of the Russian Academy of Sciences, Krasnoyarsk, Russia. Published in Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 539–548, July–August, 1996.     

Stabilizability of a class of stochastic bilinear hybrid systems

The problem of the stabilizability of stochastic nonlinear hybrid systems with a Markovian or any switching rule is considered. Using the Lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a Markovian or any switching rule is discussed and a closed-loop controller is found. Also the sufficient conditions for the exponential mean-square stabilizability for bilinear hybrid systems with any switching based on the Lie algebra approach are formulated and an open-loop controller is designed. The obtained results are illustrated by examples and simulations.     

Some estimates for the torsional rigidity of composite rods

A well‐known problem in elasticity consists in placing two linearly elastic materials (of different shear moduli) in a given plane domain Ω, so as to maximize the torsional rigidity of the resulting rod; moreover, the proportion of these materials is prescribed. Such a problem may not have a classical solution as the optimal design may contain homogenization regions, where the two materials are mixed in a microscopic scale. Then, the optimal torsional rigidity becomes difficult to compute. In this paper we give some different theoretical upper and lower bounds for the optimal torsional rigidity, and we compare them on some significant domains. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudoelasticity of Shape Memory Alloys - a 1D Material Model and Finite Element Implementation

This contribution is concerned with the formulation of a 1D-constitutive model accounting for the pseudoelastic behavior of shape memory alloys. The stress-strain-relationship is idealized by a hysteresis both in the compression as in the tension loading range. It is characterized by an upper loading path, which is to be ascribed to the transformation of the lattice to a martensitic structure. Unloading the material, a lower path is described, because of the reverse transformation into austenitic lattice. The constitutive model is based on a switching criterion which serves as a potential function for the evolution of the internal state variables. The model distinguishes between local and global variables to describe the hysteresis effects for the compression and tension range. A strain driven algorithm which captures the complete nonlinear material behavior is presented. The boundary value problem is solved for a truss element applying the finite element method. A consistent linearization of the nonlinear equations is derived. Simple examples will demonstrate the applicability of the proposed model. For future developments the usage of shape memory alloys within civil engineering structures is aimed. The advantage of the material is the very good damping behavior and the potential to overcome great strains. Both properties are distinguished to be of engineering interest. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic properties of nonlinear autoregressive Markov processes with state-dependent switching

In this paper, we consider a class of nonlinear autoregressive (AR) processes with state-dependent switching, which are two-component Markov processes. The state-dependent switching model is a nontrivial generalization of Markovian switching formulation and it includes the Markovian switching as a special case. We prove the Feller and strong Feller continuity by means of introducing auxiliary processes and making use of the Radon-Nikodym derivatives. Then, we investigate the geometric ergodicity by the Foster-Lyapunov inequality. Moreover, we establish the V-uniform ergodicity by means of introducing additional auxiliary processes and by virtue of constructing certain order-preserving couplings of the original as well as the auxiliary processes. In addition, illustrative examples are provided for demonstration.     

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)     

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.