Electroelastic Tensor Research Articles

Resonant manifestation of intrinsic nonlinearity within electroelastic micropower generators

This letter investigates the nonlinear response of a bimorph energy harvester comprised of lead zirconate titanate (PZT-5A) laminates. For near resonant excitations, we demonstrate significant intrinsic nonlinear behavior despite geometrically linear motion. Fourth order elastic and electroelastic tensor values for PZT-5A are identified following methods recently published concerning a PZT-5H bimorph. A response trend indicative of a nonlinear dissipative mechanism is discussed as well as the inadequacy of linear modeling. The PZT-5A bimorph exhibits an increased softening frequency response in comparison to PZT-5H. The results contained herein are also applicable to electroelastic sensor and actuator technologies.

Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification

We propose and experimentally validate a first-principles based model for the nonlinear piezoelectric response of an electroelastic energy harvester. The analysis herein highlights the importance of modeling inherent piezoelectric nonlinearities that are not limited to higher order elastic effects but also include nonlinear coupling to a power harvesting circuit. Furthermore, a nonlinear damping mechanism is shown to accurately restrict the amplitude and bandwidth of the frequency response. The linear piezoelectric modeling framework widely accepted for theoretical investigations is demonstrated to be a weak presumption for near-resonant excitation amplitudes as low as 0.5 g in a prefabricated bimorph whose oscillation amplitudes remain geometrically linear for the full range of experimental tests performed (never exceeding 0.25% of the cantilever overhang length). Nonlinear coefficients are identified via a nonlinear least-squares optimization algorithm that utilizes an approximate analytic solution obtained by the method of harmonic balance. For lead zirconate titanate (PZT-5H), we obtained a fourth order elastic tensor component of c1111p=−3.6673×1017 N/m2 and a fourth order electroelastic tensor value of e3111=1.7212×108 m/V.

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.

Critical Volume Fraction of Multiple Cracks for Fracture in Piezoelectric Media

This paper concerns an infinite piezoelectric solid containing multiple penny-shaped cracks subjected to a set of uniform electromechanical loads. Based on the eigenstrain formulation, the crack is treated as an inclusion by taking the electroelastic moduli of the inclusion as zero. The Mori-Tanaka theory is employed to account for the effects of crack interaction at finite concentration through the use of electroelastic Eshelby tensors. By using this theory, the averaging electroelastic field and the effective electroelastic moduli of the piezoelectric cracked body are expressed explicitly. Furthermore, an interaction energy density function is introduced to take the interaction between electromechanical loads and cracks into account. With this interaction energy density function, the critical volume fraction of multiple cracks for fracture is obtained for a simple tension, a pure shear, and a normal electric displacement, separately.The resulting critical volume fraction is a function of the geometrical dimension of the crack length, electromechanical loading, and the piezoelectric properties of the surrounding matrix.

Effective properties of polycrystalline piezoelectric ceramics

The paper focuses on the overall linear elastic, piezoelectric and dielectric response of polycrystalline piezoselectric ceramics. The effective material constants are calculated using the effective nledium approximation and single-crystal data for the individual grains. This self-consistent approach takes into account the full electromechanical interaction coupling of the grains with their surroundings in a statistical way and makes use of the rigorous treatment of the piezoelectric inclusion problem and explicit representations of the electroelastic Eshelby tensor recently published in the literature. Numerical results for the effective properties of BaTiO 3 cermaics for different microstructural arrangements of crystal orientations are presented. Although we restricted ourselves to the linear material behaviour neglecting all nonlinear effects, the calculated results show remarkable agreement with experimental data and give an interesting insight into the effect of elastic, piezoelectric and dielectric interactions on the macroscopic material behaviour.

An exact correspondence between plane piezoelectricity and generalized plane strain in elasticity

We consider an anisotropic body bounded by a cylindrical surface, which is infinitely long in the axial direction. Suppose the body is loaded in such a way that the field variables do not vary along the generators. An exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity. In particular, we show that by setting a linkage between the two sets of material constants, any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa. The assertation is true for rectilinearly, as well as for cylindrically, anisotropic solids. The equivalence is found for the most general anisotropic case, which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic (triclinic) elastic solid. A few degenerate systems are also identified. In addition, these correspondences can be extended to inhomogeneous media. Applied to composite materials or polycrystalline aggregates, they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors (and vice versa). We also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by Cherkaev, Lurie and Milton. In illustration, we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity. Lekhnitskii9s complex potential approach, together with the conformal mapping technique, are employed. General solutions of the fields inside the inhomogeneity and the matrix are obtained. The results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem.

Micromechanics determination of the effective properties of piezoelectric composites containing spatially oriented short fibers

This article presents an analytical unified method for determining the effective electroelastic properties of piezoelectric composites containing spatially oriented short fibers, which are treated as spheroidal inclusions. Both the matrix and inclusions are assumed to be linearly piezoelastic and transversely isotropic. The electroelastic tensors analogous to Eshelby tensors for elastic ellipsoidal inclusions have been obtained and also evaluated numerically for finite fiber aspect ratios. Utilizing these tensors and applying the Mori-Tanaka mean field theory to account for the interaction between inclusions and matrix, the effective electroelastic properties of the composites are expressed analytically in terms of phase properties, orientation angles, volume fraction and inhomogeneity shape. It is indicated that the proposed methods yield identical results when either a traction-electric displacement or an elastic displacement-electric field is prescribed on the compsite boundary. Finally, numerical examples are given for a BaTiO 3 PZT-5 H composite. The results show that the longitudinal and in-plane shear moduli increase with fiber length, while the other moduli, piezoelectric and dielectric constants decrease.

Micromechanics determinations of thermoelectroelastic fields and effective thermoelectroelastic moduli of piezoelectric composites

By modelling thermal expansion strain and electric field induced by a uniform temperature change as eigenfields, the eigenstrain formulation ( T. Mura, Micromechanics of Defects in Solids, 2nd revised edn., Martinus Nijhoff, 1987 ) for anisotropic inclusion problems is extended to consider the inherently anisotropic coupled behaviour of a piezoelectric composite. Utilizing the extended eigenstrain formulation, a unified explicit expression for the electroelastic tensors analogous to Eshelby tensors for elastic ellipsoidal inclusions is obtained. Particularly, closed forms for the electroelastic Eshelby tensors are presented when both the piezoelectric matrix and piezoelectric inclusions are transversely isotropic, and the shapes of the inclusions are spheroidal, elliptic cylindrical, circular cylindrical, penny-shaped, and ribbon-like. With the resulting tensors, analytical expressions for the thermoelectroelastic fields of the piezoelectric composite can then be obtained. Moreover based on the Mori-Tanaka theory and the equivalent inclusion method to account for interaction at finite concentrations of inhomogeneities, the effective thermal expansion coefficients, pyroelectric coefficients, elastic moduli, piezoelectric coefficients, and dielectric constants of the piezoelectric composite material in terms of phase properties, volume fraction, and the shape of the inhomogeneities are presented in closed forms.

An ellipsoidal inclusion or crack in orthotropic piezoelectric media

The inclusion method is developed to examine coupled electroelastic fields in an infinitely extended orthotropic piezoelectric material containing an ellipsoidal inclusion. By utilizing this method, unified expressions for both the coupled electroelastic fields within the inclusion and a set of four electroelastic tensors analogous to the Eshelby tensor for elastic inclusion problems are presented. In particular, the tensors for some practical inclusions, such as elliptical cylinder, circular cylinder, and penny shape, are derived analytically. Furthermore, by taking the electroelastic moduli of the inclusion as zero, the resulting expressions are applied to study crack problems. Three loading cases, a simple tension, a pure shear, and a normal electric displacement, are considered to examine the behavior of flat ellipsoidal cracking. Then, the critical loads of the Griffith criterion for fracture are expressed in closed forms for the electromechanical loading cases. These results could provide us with insight into how the orthotropic piezoelectric material consisting of inclusions will perform and would be helpful in understanding the damage behavior.

The electroelastic tensor of quartz determined by the resonator method

The complete electroelastic tensor of quartz has been calculated—for the first time—exclusively using the data obtained by the resonator method. Its eight independent components were obtained by the least-squares fit to 184 observations of the dc-field-induced change in the resonance frequency of the extentional mode of rods and thickness modes of plates. Simultaneously, an attempt has been made to calculate the eight independent components of the electrostrictive tensor and the only independent component of the third-order permittivity tensor of quartz. Only seven values have been obtained here; three of them are linear combinations of these quantities. A comparison made with similar tensors as calculated from the transit-time experiment [Kittinger, Tichy, and Friedel, J. Appl. Phys. 60, 1465 (1986)] yields excellent agreement for the electroelastic tensor. Differences between the corresponding tensor components are comparable with their standard errors. The comparisons available for the electrostrictive tensor are considerably less encouraging. No comparison for the third-order permittivity (disregarded in the analysis of the transit-time experiment) could be made.

Independent verification of the values of electroelastic constants of alpha quartz

The component values of the cumulative electroelastic tensor of quartz found by C. K. Hruska (Proceedings of the 31st Annual Frequency Control Symposium, June 1977, pp. 159–170; available from NTIS, Springfield, VA, as document AD A088221) by means of the resonator method are independently confirmed by G. A. Reider, E. Kittinger, and J. Tichy [J. Appl. Phys. 53, 8716 (1982)] using a time delay experiment. The disagreement of several orders of magnitude existing with other authors who also used the resonator method is explained and removed. There can be no doubt that the magnitude of the electroelastic tensor components is in the range of 1 and 10−1 N/(V m) and that the standard errors of their determined values can be as small as several percent.

Second-order electroelastic tensor of cubic KAl(SO 4) 2·12H 2O

The second-order electroelastic tensor { Z ijklmn }, which describes the variation of the elasticity tensor under the influence of an electric field in centrosymmetric crystals, has been completely determined on cubic KAl(SO 4) 2·12H 2O by measuring the electric-field-induced shift of ultrasonic resonance frequencies. The state of resonance was detected by diffraction of light through the standing ultrasonic grating formed in the state of resonance. The necessary resolution of 10 -9 for the relative frequency change was achieved employing a phase-sensitive amplification of the modulated part of the diffracted light. All main components of the second-order electroelastic tensor possess values of about -15·10 -10 NV -2. The negative sign indicates a general weakening of the elastic bonds by an electric field. A distinct anisotropy is observed in accordance with point symmetry m3.

Symmetry of effective electroelastic tensor for point group 32

The detailed symmetry properties of a fifth-rank tensor of the type ${f}_{\mathrm{mijkl}}$ which possesses interchange symmetry of the pairs $\mathrm{ij}$ with $\mathrm{kl}$ but has no interchange symmetry within these pairs is investigated for the crystal symmetry of $\ensuremath{\alpha}$-quartz [point group 32 (${D}_{3}$)]. Tensors of this type occur in the theory of the electroelastic effect. An extended matrix notation appropriate to cover this situation is proposed and used to present the results in compact form.

MEASUREMENT OF ALL COMPONENTS OF THE ELECTRO-ELASTIC TENSOR FOR α-LiIO3 CRYSTAL

The persent work discusses the method of measuring the all components of the electro-elastic tensor for the α-LiIO3 crystal (LI). By means of theory of the thickness vibrational mode for piezoelectric plates, calculating formula of independent electro-elastic constants for LI had been developed. We measured the nonpiezoelectric-coupled elastic constants c11E and c66E by method of measuring velocity with acousto-optic synchronism, Finally, the 52 data of all components of the electro-elastic tensor and some of the electromechanical coupling factors for LI are given.

Polarizing Effect with Longitudinally Vibrating ADP Rods and the Electroelastic Tensor

Polarizing Effect with Longitudinally Vibrating ADP Rods and the Electroelastic Tensor

Polorizing Effect With Alpha-Quartz Rods and the Electroelastic Tensor

Polorizing Effect With Alpha-Quartz Rods and the Electroelastic Tensor