Infinite Piezoelectric Medium Research Articles

A crack in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium

An effective method is developed and used to investigate the elastic field and the electric field of a crack in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium. The matrix is subjected to the remote antiplane shear and inplane electric field. The analytical solution is obtained using the conformal mapping and the theorem of analytic continuation. Specific solutions which are compared with existing ones are provided. The characteristics of the elastic field and electric field singularities at the crack tip are analyzed and the corresponding crack extension forces are derived.

A rigid line in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium

An effective method is developed and used to investigate the elastic field and electric field of a rigid line in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium. The matrix is subjected to the remote antiplane shear and inplane electric field. The analytical solution is obtained using the conformal mapping and the theorem of analytic continuation. Specific solutions which are compared with existing ones are provided. The characteristics of the elastic field and electric field singularities at the rigid line tip are analyzed and the extension forces on the rigid line are derived.

Linear electro-elastic analysis of a cavity or a crack in a piezoelectric material

Analytical solutions for an elliptical cylinder cavity or a crack inside an infinite piezoelectric medium under combined mechanical-electrical loadings are formulated via the Stroh formalism and well confirmed by finite element analysis. The results show that the stress and electric fields in the vicinity of the crack tip are, in general, determined by a complex vector of intensity factors. The complex vector of intensity factors may reduce to be real under certain circumstances. For a vacuum crack, the electric field inside the crack in the direction perpendicular to the crack magnifies the corresponding applied electric field by more than 1000 times. In this case, the electric field strength in the material has a finite value at the crack tip, but the electric displacement approaches infinity at the crack tip due to the piezoelectricity. The self-consistent analysis is developed to determine the deformed crack profile. The energy release rate for the cavity propagation is formulated under the condition that the ratio of the minor semi-axis to the major semi-axis of the ellipse remains unchanged. For an insulating crack, the applied electric field contributes nothing to the energy release rate when the undeformed crack profile is used, while the electric field resists crack propagation when the deformed crack profile is used. For a conducting slit crack, the energy release rate is independent of the applied electric field perpendicular to the crack, and is enhanced by the applied electric field parallel to the crack.

Electroelastic fields and effective moduli of a medium containing cavities or rigid inclusions of arbitrary shape under anti-plane mechanical and in-plane electric fields

We examine the two-dimensional problem of an infinite piezoelectric medium containing a solitary cavity or rigid inclusion of arbitrary shape, subjected to a coupled anti-plane mechanical and in-plane electric load at the remote boundary of the matrix. Conformal mapping techniques are employed to analyze the boundary value problems. Specific results are given for elliptical, polygonal and star-shape inclusions. Local fields of this type are used to estimate the overall moduli of a medium containing voids or rigid inclusions. This is accomplished with the help of an extension of Eshelby's formula which evaluates the total electric enthalpy by a particular line integral. Explicit estimates of the effective moduli are derived for dilute as well as for moderate area fractions of inclusions. The formulae depend solely on the cross area of the inclusion, area fraction and one particular coefficient of the mapping function. In addition, the stress and electric displacement singularities around the sharp corners of the inclusion are examined. The existence of uniform fields inside the inclusion is also envisaged. The present results, with appropriate modifications, apply equally well to those of thermoelectric and magnetoelectric effects.

Electromechanical analysis of defects in piezoelectric materials

In this paper, electromechanical analysis is performed based on an extended Stroh formalism which considers the constitutive law of a linear anisotropic piezoelectric medium, the kinematics of two-dimensional small deformations and static equilibrium. By this formalism, several analytical closed-form solutions are obtained for holes, cracks or interface cracks in an infinite piezoelectric medium under various loading conditions. To have a practical use for engineering analyses, the Green's functions obtained from point load condition are embedded into the usual boundary integral equations. Thus, a special boundary element is developed to deal with the problems with finite boundaries.

Fields near elliptical crack tip in piezoelectric ceramics

The field equations of three-dimensional problems in a transversely isotropic piezoelectric medium become four simultaneous partial differential equations of second order, in which the displacement components and electric potential are taken as the essential unknowns. In this paper the general solutions of the above mentioned equations, expressed by the newly introduced potential functions, are deduced. Therefore, the obtained general solutions are applied to solve a problem in an infinite piezoelectric medium with an elliptical crack on whose surfaces uniformly distributed mechanical and electric loads are exerted, and the crack surfaces are assumed to be parallel to the material plane of isotropy. The analytical expressions of the stress and electric displacement fields near the crack tip show an extremely complicated coupling relationship between the mechanical and electric variables. To illustrate the results acquired in this paper an example for a kind of piezoelectric ceramics applied in engineering is computed finally.

Boundary integral formulations for three-dimensional anisotropic piezoelectric solids

A boundary integral equation method is applied to the solutions of three dimensional piezoelectric solids. Based on the reciprocal relations, a pair of boundary integral formulae were formulated for evaluation of the fields in the medium. The Green's functions and their first partial derivatives employed in the formulations are evaluated numerically from the line integral solutions derived from the Fourier transform. By constructing some augmented matrices, we show that the topic can be treated systematically as that in the uncoupled elastic and dielectric problems. In illustration, we present results for the internal fields of a spherical cavity in an infinite piezoelectric medium loaded by a uniform traction on its boundary. Two piezoelectric ceramics, PZT-6B and gallium arsenide, are considered in the calculations. Some comparisons are made with solutions of purely elastic solids and with our recent calculations based on the finite element method.

An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities

A set of four tensors corresponding to Eshelby’s tensor in elasticity are obtained for an ellipsoidal inclusion embedded in an infinite piezoelectric medium. These tensors, which describe the elastic, piezoelectric, and dielectric constraint of the matrix, are obtained from W. F. Deeg’s solution to inclusion and inhomogeneity problems in piezoelectric solids. These tensors are then used as the backbone in the development of a micromechanics theory to predict the effective elastic, dielectric, and piezoelectric moduli of particle and fibre reinforced composite materials. The effects of interaction among inhomogeneities at finite concentrations are approximated through the Mori-Tanaka mean field approach. This approach, although widely utilized in the study of uncoupled elastic and dielectric behaviour, has not before been applied to the study of coupled behaviour. To help ensure confidence in the theory, the analytical predictions are proven to be self-consistent, diagonally symmetric, and to exhibit the correct behaviour in the low and high concentration limits. Finally, numerical results are presented to illustrate the effects of the concentration, shape, and material properties of the reinforcement on the effective properties of piezoelectric composites and analytical predictions are shown to result in good agreement with existing experimental data.

Effect of electric field on fracture of piezoelectric ceramics

Closed form solutions for all three modes of fracture for an infinite piezoelectric medium containing a center crack subjected to a combined mechanical and electrical loading were obtained. The explicit mechanical and electrical fields near the crack tip were derived, from which the strain energy release rate and the total potential energy release rate were obtained by using the crack closure integral. The suitability in using the stress intensity factor, the total energy release rate, or the mechanical strain energy release rate as the fracture criterion was discussed.

Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites

The dilute, self-consistent, Mori-Tanaka and differential micromechanics theories are extended to consider the coupled electroelastic behavior of piezoelectric composite materials. The application of each theory is based on Deeg's (1980, unpublished) rigorous three-dimensional electroelastic solution of an ellipsoidal inclusion in an infinite piezoelectric medium. Each micro-mechanics theory is implemented through a matrix formulation in which the effective electroelastic moduli are conveniently represented by a 9 × 9 matrix. As in the corresponding uncoupled elastic and electric behavior, the dilute and Mori-Tanaka schemes return explicit estimates for the effective electroelastic moduli. The self consistent method, however, returns an implicit nonlinear algebraic matrix equation for the effective electroelastic moduli. The differential scheme formally results in a set of 81 coupled nonlinear ordinary differential equations for the effective electroelastic moduli. In general, recourse to a numerical scheme is required for the self-consistent and differential theories. Numerical results are presented to illustrate the performance of each model for some typical composite microstructures and the models are compared in light of existing experimental data.

Linear electro-elastic fracture mechanics of piezoelectric materials

The concepts of linear elastic fracture mechanics, generalized to treat piezoelectric effects, are employed to study the influence of the electrical fields on the fracture behavior of piezoelectric materials. The method of distributed dislocations and electric dipoles, already existing in the literature, is used to calculate the electro-elastic fields and the energy-release rate for a finite crack embedded in an infinite piezoelectric medium which is subjected to both mechanical and electric loads. The energy-release rate expressions show that the electric fields generally tend to slow the crack growth. It is shown that the stress intensity factor criterion and the energy-release rate criterion differ when the energetics of the electric field is taken into account. The study of crack tip singular stress field yields a possible explanation for experimentally observed crack skewing in the presence of a strong electric field.

Wave propagation in piezoelectric medium of hexagonal symmetry

The wave propagation at large distances from a source of disturbance (isolated harmonic electric charge or body force of fixed frequency) in an infinite piezoelectric medium belonging to classes (6), (6 m m) or ceramic (α m) and (6 2 2) is discussed by means of asymptotic evaluation (at large distances) of Fourier integrals. Numerical results are given for the (6 m m) crystal class using the constants of cadmium selenide crystal.