Thermodynamic Modeling of Hysteresis Effects in Piezoceramics for Application to Smart Structures
01 Jan 2008-AIAA Journal (American Institute of Aeronautics and Astronautics (AIAA))-Vol. 46, Iss: 1, pp 280-284
TL;DR: In this paper, the effect of the amplitude of the electric field on the hysteresis loop has been studied experimentally by Nalwa et al. under cyclic variation of the applied electric field, which leads to an interesting variation of strain with respect to electric field (E), and it is denoted as butterfly loop.
Abstract: W ITH the development of piezoceramic sensors and actuators of varying shapes and sizes for use in structural applications, the field of smart structures has emerged as an area of research of great importance [1]. The mechanical, thermal, and electrical behavior of piezoceramics has been studied extensively by physicists and material scientists [2–9]. The introduction of these materials in structural applications has created a necessity to review the traditional structural modeling and analysis. Under cyclic variation of the applied electric field, piezomaterials exhibit polarization-electric (P-E) field hysteretic losses, as shown in Fig. 1. The points indicated by the symbols Ps, Pr, and EC represent saturation polarization, remnant polarization, and the coercive electric field, respectively. The saturation polarization Ps corresponds to the value of maximum polarization, which shows negligible change with further increase in electric field. Remnant polarization Pr is the value of polarization when the electric field becomes zero. Coercive electric fieldEC corresponds to the points of zero polarization. The observed phenomenon is due to the delay in polarization switching with variation in electric field. P-E hysteresis effect leads to an interesting variation of strainwith respect to electric field ( E), and it is denoted as butterfly loop. The hysteresiswill be affected by various parameters such as temperature, amplitude of oscillating electric field, frequency of oscillation, and external mechanical preloading. The effect of the amplitude of the electric field on the hysteresis loop has been studied experimentally by Nalwa [2]. It is observed that the remnant polarization and coercive electric field are functions of amplitude of the electric field. With a decrease in amplitude of the electric field, there is a reduction in the values of maximum polarization, remnant polarization, and coercive electric field. Viehland and Chen [3] experimentally studied the effects of frequency of oscillation of the electric field on the hysteresis loop. From the experiments, it was observed that if the amplitude of the electric field is above the coercive electric field ECmax corresponding to the case with saturation polarization, then the dissipation energy increases with an increase in frequency, whereas if the amplitude of the electric field is belowECmax, then the dissipation energy increases with a decrease in frequency. The effect of mechanical preloading on hysteresis has been studied experimentally by Arndt et al. [4]. It is observed that when a compressive mechanical preloading is applied parallel to the electric field, the reduction in polarization is found to be higher than that corresponding to the case with mechanical loading applied perpendicular to the electrical field. With the application of compressive mechanical preloading, the remnant polarization and coercive field show a reduction in magnitude. Mathematical modeling of hysteresis was approached at two different levels: 1) at the microscopic level and 2) at the macroscopic level of the piezomaterial. The microscopic models of hysteresis can be categorized as 1) polarization switching based on the Eshelby inclusion model [10], 2) crystal-plasticity-based nonlinear switching models [11], 3) free-energy-based domain-switching model [12], and 4) dipole–dipole interaction models with threshold-switching energy given by the time-dependent Ginzburg–Landau model [13,14] or the Landau–Devonshire model [15,16]. Macroscopic models can be categorized as either empirical models or thermodynamically consistent models. Empirical models are based on either introducing an additional variable in the constitutive relations [17], by representing the P-E curve by a tanhyperbolic function [18], or by using the Presaich model [19,20]. Bassiouny et al. [21–24] developed a thermodynamic phenomenological model for capturing the electromechanical hysteresis effects based on the work-hardening plasticity model. A similar approach was followed by McMeeking and Landis [25] in modeling domain switching in ferroelectric materials. A model based on extended irreversible thermodynamics was proposed by Lu and Hanagud [26]. Presented as Paper 1743 at the 15-th AIAA/ASME/AHS Adaptive Structures Conference, Honolulu, HI, 23–26 April 2007; received 1 May 2007; accepted for publication 25 August 2007. Copyright © 2007 by V. L. Sateesh, C. S. Upadhyay, and C. Venkatesan. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/08 $10.00 in correspondence with the CCC. ∗Graduate Student, Department of Aerospace Engineering. Student Member AIAA. Associate Professor, Department of Aerospace Engineering. Pandit Ramachandra Dwivedi Chair Professor, Department of Aerospace Engineering. Senior Member AIAA. AIAA JOURNAL Vol. 46, No. 1, January 2008
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TL;DR: In this paper, the second-order nonlinear constitutive equations are used in the variational principle approach to develop a nonlinear finite element (FE) model for piezoelectric laminated composite plates and shells.
Abstract: In this article, we focus on static finite element (FE) simulation of piezoelectric laminated composite plates and shells, considering the nonlinear constitutive behavior of piezoelectric materials under large applied electric fields. Under the assumptions of small strains and large electric fields, the second-order nonlinear constitutive equations are used in the variational principle approach, to develop a nonlinear FE model. Numerical simulations are performed to study the effect of material nonlinearity for piezoelectric bimorph and laminated composite plates as well as cylindrical shells. In comparison to the experimental investigations existing in the literature, the results predicted by the present model agree very well. The importance of the present nonlinear model is highlighted especially in large applied electric fields, and it is shown that the difference between the results simulated by linear and nonlinear constitutive FE models cannot be omitted.
23 citations
TL;DR: In this paper, a micromechanically motivated model is embedded into an electromechanical coupled finite element formulation in which each grain is represented by a single finite element and the initial dipole directions are assumed to be randomly oriented to mimic the virgin state of the unpoled ferroelectric polycrystal.
Abstract: The aim of this paper is to capture the grain boundary effects taking into consideration the nonlinear dissipative effects of ferroelectric polycrystals based on firm thermodynamic principles. The developed micromechanically motivated model is embedded into an electromechanically coupled finite element formulation in which each grain is represented by a single finite element. Initial dipole directions are assumed to be randomly oriented to mimic the virgin state of the unpoled ferroelectric polycrystal. An energy-based criterion using Gibbs free energy is adopted for the initiation of the domain switching process. The key aspect of the proposed model is the incorporation of effects of the constraint imposed by the surrounding grains on a switching grain. This is accomplished by the inclusion of an additional term in the domain switching criterion that is related to the gradient of the driving forces at the boundary of the grains. To study the overall bulk ceramics behavior, a simple volume-averaging technique is adopted. It turns out that the simulations based on the developed finite element formulation with grain boundary effects are consistent with the experimental data reported in the literature.
16 citations
TL;DR: In this article, an attempt is made to study these (P-E) nonlinear effects on the static response of laminated composite plates with piezo actuators and to find the most effective piezo lay-up and ply orientation which gives the maximum deflections.
Abstract: Polarization–electric-field (P–E) interaction results in rendering the stress tensor non-symmetric and in a nonlinear force term in the equilibrium equation. In this paper, an attempt is made to study these (P–E) nonlinear effects on the static response of laminated composite plates with piezo actuators. Further, this paper also focuses on finding the most effective piezo lay-up and ply orientation which gives the maximum deflections. Four different piezo lay-up configurations and three ply orientations are considered. It has been observed from the study that width-wise strips show more transverse bending and twisting. However, full length piezo layers show maximum longitudinal bending. The results of nonlinear analysis show a more considerable softening trend in deformations than that of the linear analysis in the case of longitudinal bending and twisting. In the case of transverse bending this nonlinear effect shows a hardening trend. Further, it has been observed that the influence of P–E nonlinearity depends on the stiffness of the core material, the geometric arrangement of piezo patches, the boundary conditions and the actuation voltage.
6 citations
TL;DR: In this paper, a higher order finite element model has been developed for the analysis of debonding in a smart cantilever beam, where the debonding has been incorporated at the interfaces between piezo patches and the core.
Abstract: Using basic electro-elastic formulation and variational formulation, a higher order finite element model has been developed for the analysis of debonding in a smart cantilever beam. Full length piezo patch embedded at the top and bottom of the aluminium core has been assumed to be de-bonded. The debonding has been incorporated at the interfaces between piezo patches and the core, at the mid span of the beam for one third length of the beam. The effect of debonding in sensing mode has been analysed by presenting the induced potential, axial displacement, axial/transverse electric field and stresses for fully bonded and de-bonded smart cantilever beam. The variation in electric potential, electric field, axial displacement/strain/stress and shear strain/stress observed in case of debonding demonstrates that the mechanics of debonding is complex coupled electro-mechanical behaviour. In the de-bonded beam, the induced potential at the free piezo surface and at the interfaces shows a sinusoidal variation from root to the tip as compared to the linear variation in bonded beam. This is attributed to the non-linear bending moment variation from root to the tip in case of de-bonded beam. The maximum stress in debonding increases nearly 1.5 times to that of bonded beam sensing at various locations.
2 citations
TL;DR: In this article, a layer-by-layer finite element formulation is developed for both static and dynamic analyses of smart plates including hysteresis effects, and linear and nonlinear analyses are carried out to study the effect of nonlinearity due to polarization-electric field interaction on the response of the smart plates.
Abstract: The coupled electrothermoelastic constitutive relations, representing the behavior of piezo materials, are shown to be nonlinear when the polarization―electric field interaction effects are properly accounted for. Under static condition, the polarization―electric field nonlinearities correspond to a distributed body force and body moment. For time varying situations, polarization―electric field interaction exhibits a hysteresis effect. Using the nonlinear constitutive relations, a layer-by-layer finite element formulation is developed for both static and dynamic analyses of smart plates including hysteresis effects. Linear and nonlinear analyses are carried out to study the effect of nonlinearity due to polarization―electric field interaction on the response of the smart plates. To validate the present formulation, the results of static analysis of a smart plate are compared with experimental data available in literature. It is observed that the P-E hysteresis effects show a friction-type damping in the dynamic response of the smart plate.
1 citations
Cites background or methods from "Thermodynamic Modeling of Hysteresi..."
...The material considered for the study is PZT-5H and values of all material constants used for generating the hysteresis loops are listed in the Tables 1–3 [15]....
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...Using the expression for and by neglecting temperature effects, the set of constitutive relations representing electroelastic process including electric hysteresis effects can be written as [15]...
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...Nonlinear Constitutive Relations and Governing Equations For obtaining the constitutive relations, the free energy is expressed as a quadratic function of all state variables and quartic functions of strain and electric field [15]:...
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...[15] have derived the electrothermoelastic constitutive relations for the piezo materials including the hysteresis effects by following fundamental principles of physics....
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References
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TL;DR: In this paper, a computer simulation model based on the time-dependent Ginsburg-Landau equations was used to simulate the formation and evolution of a cubic-to-tetragonal ferroelectric phase transition.
Abstract: Three-dimensional (3-D) computer simulations of ferroelectric domain formation and evolution were performed, using a computer simulation model based on the time-dependent Ginsburg-Landau equations. A cubic-to-tetragonal ferroelectric phase transition is considered. It is shown that the initial stage of the transition during the annealing of a quenched cubic paraelectric phase involves the nucleation and growth of the ferroelectric domains, followed by the domain coarsening leading to the formation of 90° and 180° domain structures. Part of the 3-D results reported here confirm our conclusions made earlier for the two-dimensional (2-D) case, namely, the nonlocal elastic interactions are critical to the formation of twin structure and the dipole-dipole interactions are responsible for the head-to-tail arrangements of dipoles at twin boundaries. In contrast to our previous work, and others; the effect of the depolarization energy was explicitly incorporated into the simulation model. It is found that when there are no surface charges to compensate the Lorentz field due to the polarization charges, and if the system is mechanically clamped, both 90° and 180° domains are thermodynamically stable.
203 citations
"Thermodynamic Modeling of Hysteresi..." refers methods in this paper
...The microscopic models of hysteresis can be categorized as 1) polarization switching based on the Eshelby inclusion model [10], 2) crystal-plasticity-based nonlinear switching models [11], 3) free-energy-based domain-switching model [12], and 4) dipole–dipole interaction models with threshold-switching energy given by the time-dependent Ginzburg–Landau model [13,14] or the Landau–Devonshire model [15,16]....
[...]
TL;DR: In this paper, a polarization switching model for polycrystalline ferroelectric ceramics was developed, in which a single ferro-electric crystallite in a ceramic, which is subjected to an electric field and/or a stress, undergoes a complete polarization change and a corresponding strain change if the resulting reduction in potential energy exceeds a critical value per unit volume of switching material.
Abstract: A polarization switching model for polycrystalline ferroelectric ceramics has been developed. It is assumed that a single ferroelectric crystallite in a ceramic, which is subjected to an electric field and/or a stress, undergoes a complete polarization change and a corresponding strain change if the resulting reduction in potential energy exceeds a critical value per unit volume of switching material. The crystallite’s switch causes a change in the interaction of its field and stress with the surrounding crystallites, which is modeled by the Eshelby inclusion method to provide a mean field estimate of the effect. Thus the model accounts for the effects of the mean electric and stress fields arising from the constraints presented by surrounding crystallites as well as the externally applied mechanical and electrical loads. The switching response of the ceramic polycrystal is obtained by averaging over the behavior of a large number of randomly oriented crystallites. The model, along with the linear dielect...
191 citations
"Thermodynamic Modeling of Hysteresi..." refers methods in this paper
...The microscopic models of hysteresis can be categorized as 1) polarization switching based on the Eshelby inclusion model [10], 2) crystal-plasticity-based nonlinear switching models [11], 3) free-energy-based domain-switching model [12], and 4) dipole–dipole interaction models with threshold-switching energy given by the time-dependent Ginzburg–Landau model [13,14] or the Landau–Devonshire model [15,16]....
[...]
TL;DR: In this paper, the authors examined the domain patterns and macroscopic behavior of single crystals of ferroelectric material using a theory based on energy minimization, and identified a low-energy path for domain switching and a novel configuration that yields very large electrostriction.
Abstract: This paper examines the domain patterns and the macroscopic behaviour of single crystals of ferroelectric material using a theory based on energy minimization. A low-energy path is identified for domain switching and a novel configuration that yields very large electrostriction is identified.
184 citations
"Thermodynamic Modeling of Hysteresi..." refers methods in this paper
...The microscopic models of hysteresis can be categorized as 1) polarization switching based on the Eshelby inclusion model [10], 2) crystal-plasticity-based nonlinear switching models [11], 3) free-energy-based domain-switching model [12], and 4) dipole–dipole interaction models with threshold-switching energy given by the time-dependent Ginzburg–Landau model [13,14] or the Landau–Devonshire model [15,16]....
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TL;DR: In this article, a phenomenological constitutive law for ferroelectric switching due to multi-axial mechanical and electrical loading of a polycrystalline material is developed, which is based on kinematic hardening plasticity theory and has a switching surface in the space of mechanical stress and electric field.
Abstract: A phenomenological constitutive law for ferroelectric switching due to multi-axial mechanical and electrical loading of a polycrystalline material is developed. The framework of the law is based on kinematic hardening plasticity theory and has a switching surface in the space of mechanical stress and electric field that determines when non-linear response is possible. The size and shape of the switching surface in a modified electric field space remains fixed during non-linear behavior but its center moves around and thus is controlled by a kinematical hardening process. In general, the remanent polarization and the remanent strain are used as the internal variables that control how the center of the switching surface moves. However, the form presented in this paper has a one-to-one relationship between the remanent strain and the remanent polarization, simplifying the constitutive law and allowing remanent polarization to be used as the only internal variable controlling the kinematic effects. The constitutive law successfully reproduces hysteresis and butterfly loops for ferroelectric ceramics. The hysteresis and butterfly loops respond appropriately to the application of a fixed compressive stress parallel to the electric field. In addition, the law successfully handles remanent polarization rotation due to the application of electric field at an angle to the polarization direction.
179 citations
"Thermodynamic Modeling of Hysteresi..." refers methods in this paper
...A similar approach was followed by McMeeking and Landis [25] in modeling domain switching in ferroelectric materials....
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TL;DR: In this paper, a new concept of polarization reversal and domain dynamics in modified ferroelectrics is presented, where clusters with a reversed polarization within ferroelectric domains are nucleated under applied field in the vicinity of random fields around quenched defects.
Abstract: A new concept of polarization reversal and domain dynamics in modified ferroelectrics is presented. Accordingly, clusters with a reversed polarization within ferroelectric domains are nucleated under applied field in the vicinity of random fields around quenched defects. The random fields significantly lower the activation barrier required for nucleation, thus controlling the kinetics of polarization reversal and domain dynamics. In this model, polarization switching occurs by the breakdown of ferroelectric long-range order, through a polar cluster state.
176 citations