Computational modeling of rate-dependent domain switching in piezoelectric materials

In this contribution a micromechanically motivated model for rate-dependent switching effects in piezoelectric materials is developed. The proposed framework is embedded into a three-dimensional finite element setting whereby each element is assumed to represent an individual grain. Related dipole (polarization) directions are thereby initially randomly oriented at the element level to realistically capture the originally un-poled state of grains in the bulk ceramics. The onset of domain switching processes is based on a representative energy criterion and combined with a linear kinetics theory accounting for time-dependent propagation of domain walls during switching processes. In addition, grain boundary effects are incorporated by making use of a macromechanically motivated probabilistic approach. Standard volume-averaging techniques with respect to the response on individual grains in the bulk ceramics are later on applied to obtain representative hysteresis and butterfly curves under macroscopically uniaxial loading conditions at different loading frequencies. It turns out that the simulations based on the developed finite element formulation nicely match experimental data reported in the literature.     

A multi-surface model for ferroelectric ceramics - application to cyclic electric loading with changing maximum amplitude

Depending on the maximum amplitude of externally applied cyclic electric fields, ferroelectric ceramics show minor or major hysteresis. The materials also show asymmetric butterfly hysteresis in a prepoled material. Aiming at capturing these behaviour in a phenomenological constitutive model, a multi-surface modelling approach for ferroelectrics is introduced. In this paper, with the note on the motivation for a multi-surface model related to the results of new experimental investigations and also to experimental data reported in the literature, the constitutive relation for a rate dependent multi-surface ferroelectric model is developed. Following this, a brief graphical illustration shows how this model captures the objective phenomena. Consequently, the numerical implementation of the model to capture experimental results is demonstrated. Finally, the performance of this model to represent behaviour of decaying polarisation offset of electrically fatigued specimen is shown.     

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)     

Non-linear electromechanical response of 1–3 type piezocomposites

Domain switching in piezoelectric materials is caused by external loads such as electric field and stress that leads to non-linear behaviour. A study is carried out to compare the non-linear behaviour of 1–3 piezocomposites with different volume fractions and bulk piezoceramics. Experiments are conducted to measure the electrical displacement and strain on piezocomposites and bulk ceramics under high cyclic electrical loading and constant compressive prestress. A thermodynamically consistent uni-axial framework is developed to predict the nonlinear behaviour by combining the phenomenological and micromechanical techniques. Volume fractions of three distinct uni-axial variants (instead of six variants) are used as internal variables to describe the microscopic state of the material. In this model, the grain boundary effects are taken into account by introducing the back fields (electric field and stress) as non-linear kinematic hardening functions. An analytical model based on equivalent layered approach is used to calculate effective properties such as elastic, piezoelectric, and dielectric constants for different volume fractions of piezocomposites. The predicted effective properties are incorporated in the proposed uni-axial model and the dielectric hysteresis (electrical displacement versus electric field) as well as butterfly curves (strain versus electric field) are simulated. Comparison between the experiments and simulations show that this model can reproduce the characteristics of non-linear response. It is observed that the variation in fiber volume fraction and compressive stress has a significant influence on the response of the 1–3 piezocomposites.     

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)     

Micromechanically motivated finite element model for ferroelectric ceramics

Ferroelectric ceramics exhibit significant coupled electromechanical phenomena that have been widely employed in sensor and actuator applications. In regular finite element models dealing with electromechanical plane problems, each grain needs to be subdiscretized by many triangular or quadrilateral elements for required accuracy. This problem can be overcome by a polygonal finite element approach where each grain is modelled by a single finite element without compromising on the results. In this paper, a polygonal finite element approach has been employed to understand the anisotropic response of the ferroelectric ceramics in their piezoelectric region. (© 2009 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)     

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)