A line dislocation interacting with a semi-infinite crack in piezoelectric solid

Closed-form analytical solutions are presented for the physical problem of a semi-infinite crack interacting with a line dislocation under the loading of a line force and a line charge in two-dimensional infinite anisotropic piezoelectric medium. The crack can be a conventional Griffith crack or an anti-crack (a rigid line inhomogeneity). Using the extended Stroh formalism and perturbation technique, the explicit expressions of the field intensity factors and the image force on the dislocation are computed as functions of dislocation location and material constants. The results are discussed and compared with those from special cases existed in the literature. The analytical solutions obtained can be applied to studying interacting cracks and crack branching problems in piezoelectric solids.     

Analytical Full-field Solutions of a Piezoelectric Layered Half-plane Subjected to Generalized Loadings

The two-dimensional problem of a planar transversely isotropic piezoelectric layered half-plane subjected to generalized line forces and edge dislocations in the layer is analyzed by using the Fourier-transform method and the series expansion technique. The full-field solutions for displacements, stresses, electrical displacements and electric fields are expressed in explicit closed forms. The complete solutions consist only of the simplest solutions for an infinite piezoelectric medium with applied loadings. It is shown in this study that the physical meaning of this solution is the image method. The explicit solutions include Green's function for originally applied loadings in an infinite piezoelectric medium and the remaining terms are image singularities which are induced to satisfy free surface and interface continuity conditions. The mathematical method used in this study provides an automatic determination for the locations and magnitudes of all image singularities. The locations and magnitudes of image singularities are dependent on the piezoelectric material constants of the layered half-plane and the location of the applied loading. With the aid of the generalized Peach-Koehler formula, the image forces acting on dislocations are derived from the full-field solutions of the generalized stresses. Numerical results for the full-field distributions of stresses and electric fields in the piezoelectric layered half-plane and image forces for edge dislocations are presented based on the available analytical solutions.     

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.This work was supported by the National Science Council, Taiwan, under contract NSC 83-0410-E006-041.     

Derivatives of Green’s functions in piezoelectric media and their application in dislocation dynamics

Three-dimensional extended Green’s functions and their high-order derivatives in general anisotropic piezoelectric materials are derived and expressed in integral forms. They can be evaluated directly by the Gaussian numerical integration method. The extended Green’s functions and their derivatives by the present method have accuracy and computational efficiency. Using the extended Green’s functions, the stress field induced by an arbitrary dislocation in an anisotropic piezoelectric medium, is obtained and expressed as a line integral around the dislocation.     

New transmission line analogy applied to single and multilayered piezoelectric transducers

It is shown that a piezoelectric element vibrating in an extensional or shear mode can be modeled rigorously by systematic use of the transmission line analogy and the superposition theorem. A schematic representation of such an element which is in a way more intuitive than others representations is introduced. The stresses on the electroded faces are considered as sources of stress applied at the two ends of an acoustic transmission line, since the acoustical perturbations may be considered as originating on these faces. Using transmission line theory, a complete set of expressions is found for electrical impedance, acoustic stresses, and velocities. Computed results are exactly the same as those given by the classical method, even if the computation sequence is almost entirely different. An intuitive graphical model for a piezoelectric element is proposed. It is also shown that the acoustic velocities on opposite faces of an asymmetrical loaded piezoelectric plate are exactly equal at the antiresonance frequency when internal losses are neglected. The programs developed can be used efficiently for practical multilayered transducer design.     

2D electro-elastic fields in a piezoelectric layer-substrate structure

The coupled electro-elastic fields are found in the piezoelectric layer-substrate structure of unrestricted anisotropy containing straight-line sources in the interior or at the surface of the structure. The line sources in the interior are parallel to the surface, which is supposed to be free of traction or clamped and, electrically, metallized, free of charge or adjoined to an isotropic dielectric medium. The boundary problems of prescribed distributions of loads, displacements, electric potential or charge density at the surface are also considered. All obtained solutions are presented in a form of convergent Fourier integrals. The integrands are implicitly expressed in terms of the eigenvalues and the eigenvectors of the generalized Stroh matrix.     

Antiplane electro‐mechanical field of a piezoelectric finite wedge under a pair of concentrated forces and free charges

Abstract

This paper presents general antiplane electro‐mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as traction free and insulated. Using finite Mellin transform methods, the stress and electrical displacement in all fields of the piezoelectric finite wedge are derived analytically. Singularity orders and intensity factors of stress and electrical displacement can be obtained too. After being reduced to a problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.     

Stator design of a new type of spherical piezoelectric motor

The stator design of a new type of spherical motor driven by piezoelectric actuators is developed. A curved piezoelectric actuator is designed to attach to the spherical surface. A series of the curved piezoelectric actuators is laid in a line around a spherical surface. By applying an appropriate voltage signal with phase difference on neighboring actuators, a traveling wave is generated on the hemispherical shell. Each set of curved piezoelectric actuators is designed to provide motion with a single degree-of-freedom (DOF). With two or three sets of the piezoelectric actuators constructed to be mutually perpendicular, the motor can provide 2-DOF or 3-DOF motion. Stator design and analysis and experiment for the 1-, 2-, and 3-DOF conditions are presented in this article. Analytical calculation and experiment results of several fundamental characteristics of the stator are in good agreement. Performance evaluation of rotation speed and torque of the stator and some implementation problems are also addressed.     

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

Summary 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.     

Electrostatic tractions at crack faces and their influence on the fracture mechanics of piezoelectrics

The influence of electrostatic tractions acting upon crack faces on the fracture mechanical quantities in piezoelectric materials under electromechanical loading is investigated. The physical background are the mechanical and dielectric equilibria at an interface between two dielectric domains and related mechanical stresses. The model is applied to a crack problem, where a dielectric interface exists between the solid material and the insulating crack medium. The analytical solution for a crack in an infinite piezoelectric body accounting for intrinsic charges and electrostatic stresses on the crack faces gives insight into the influence of crack boundary conditions on the field intensity factors. Varying loading conditions and the dielectric permittivity of the flaw yields a parameter range in which induced crack surface tractions are relevant. Then, the calculation of the J-integral for thermodynamically consistent crack boundary conditions is discussed. The line integral along the crack faces is replaced by a simple jump term. This approach comes out to be exact only for a simplified model of the electrostatic tractions.     

Controlled-source analogous circuits and SPICE models forpiezoelectric transducers

Transmission line analogous circuits for piezoelectric transducers are developed which employ controlled sources rather than the traditional transformer to model the coupling between the electrical and the mechanical systems. A novel method is used to derive each model that consists of adding a term that is equal to zero to one of the device electromechanical equations. When this is done, it is shown that the equations can be cast into the form of the familiar telegraphist's equations for the voltage and current on an electrical transmission line. The circuits are derived for both the thickness-mode piezoelectric transducer and the side-electrode bar piezoelectric transducer. SPICE models of the analogous circuits are presented and an example simulation is given     

Green’s functions of a half-infinite piezoelectric body: Exact solutions

Summary. Greens functions of a half-infinite piezoelectric space play an important role in electroelastic analyses of piezoelectric media. However, almost all works available on the topic are based on the assumption that the normal component of the electric displacement is zero on the surface of the piezoelectric solid, neglecting the effect of polarized surface charge. In the present work, we develop an exact solution for the Greens functions of a half-infinite piezoelectric solid by means of the Stroh formalism. The solution is based on using the exact electric boundary conditions at the interface between the solid and the air medium. First, Greens function for an arbitrary line load in the solid is derived taking into account the effect of polarized charge at the interface, and then the surface Greens function for a surface load is obtained as a special example. Finally, by using the superposition principle, a general expression for the polarized charge distribution on the surface of the piezoelectric solid is presented when an arbitrarily distributed force is exerted on the boundary. It is shown that the normal component of the electric displacement on the solid surface is not zero and it is dependent on the applied loads and the electro-elastic constants of the piezoelectric material and air.     

Effect of finite cracking on the electromechanical coupling properties of piezoelectric materials

Cracks and porosities inside the piezoelectric materials can weaken the electromechanical coupling effect, and hence influence the electromechanical coupling behavior of piezoelectric materials considerably. This paper studies the effect of internal cracking on the effective properties of piezoelectric media. It focuses on the piezoelectric medium of finite size with finite crack. The mechanical and electric fields in the piezoelectric material and the crack are formulated by singular integral method. Effects of crack size, medium border, and electric permeability of the crack on the overall electromechanical properties of the piezoelectric material are obtained and displayed graphically. In addition, the crack tip coupling electromechanical field intensity factors are also presented as they are not available in open literature for a finite crack in a finite piezoelectric media.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

Mode-III crack problems in a cubic piezoelectric medium

Mode-III crack problems in a cubic piezoelectric medium are investigated. Electromechanical fields near the crack tip are derived, and explicit expressions are obtained. It is shown that the asymptotic stress and electric fields are characterized by the inverse square root of the distance from the tip with at most two independent coefficients. The energy release rate associated with the crack is determined, and comparison is made with the counterpart in materials of hexagonal symmetry. The fundamental problem concerning a screw dislocation which is accompanied by other line singularities of different physical nature is addressed. A class of crack problems is solved by reducing it to the standard Riemann–Hilbert problem in complex function theory. It is concluded that under certain conditions, for the same crack configuration and loading, the stress intensity factors found for isotropic elastic materials are applicable to cubic piezoelectric media. Finally, three crack problems are explicitly solved.     

The Coulombic traction on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials

This paper deals with the Coulombic traction usually neglected, but inherently acting, on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials. The dielectric material phase is treated as a special kind of piezoelectric material with a little piezoelectricity, whereas the metal phase is treated as another special kind of piezoelectric material with an extremely large permittivity and an extremely small piezoelectricity. The permittivity of the medium inside the crack gap is accounted for either. The normal electric displacement component and the Coulombic traction on the crack surfaces are unknown, and are determined from a cubic equation deduced from the extended Stroh formula. Numerical results for the Coulombic traction in both kinds of bimaterials reveal that in most cases its magnitude is remarkable and cannot be entirely neglected when the applied electric field is higher. It is concluded that in most cases the Coulombic traction yields significant influence on the effective stress intensity factor at the crack tip and may influence the fracture behavior in such kinds of bimaterials. As compared to homogenous piezoelectric materials, the metal phase always decreases the Coulombic traction, whereas the dielectric phase decreases it under the negative electric field and increases it under the positive electric field. In all cases, BaTiO₃ always yields a much larger Coulombic traction than PZT-4.     

压电螺型位错与椭圆夹杂界面导电刚性线的干涉效应

研究了无限大压电基体材料中压电螺型位错与含界面导电刚性线椭圆夹杂的电弹耦合干涉问题。运用求解复杂多连通域问题的复变函数方法,获得了椭圆夹杂和基体区域复势函数以及电弹性场的精确级数形式解答。利用广义Peach-Koehler公式导出作用于压电螺型位错上的位错力公式。主要讨论了刚性线几何尺寸和椭圆曲率对位错力的影响规律。分析结果表明:界面刚性线排斥基体中的位错,对靠近椭圆夹杂界面的螺型位错的运动和平衡位置有重要的影响。当刚性线的长度达到临界值,界面刚性线的存在会改变螺型位错与压电椭圆夹杂的干涉规律。椭圆夹杂的压缩系数变大,刚性线尺寸对位错力的影响也越大。     

Enhanced piezoelectricity and stretchability in energy harvesting devices fabricated from buckled PZT ribbons

The development of a method for integrating highly efficient energy conversion materials onto soft, biocompatible substrates could yield breakthroughs in implantable or wearable energy harvesting systems. Of particular interest are devices which can conform to irregular, curved surfaces, and operate in vital environments that may involve both flexing and stretching modes. Previous studies have shown significant advances in the integration of highly efficient piezoelectric nanocrystals on flexible and bendable substrates. Yet, such inorganic nanomaterials are mechanically incompatible with the extreme elasticity of elastomeric substrates. Here, we present a novel strategy for overcoming these limitations, by generating wavy piezoelectric ribbons on silicone rubber. Our results show that the amplitudes in the waves accommodate order-of-magnitude increases in maximum tensile strain without fracture. Further, local probing of the buckled ribbons reveals an enhancement in the piezoelectric effect of up to 70%, thus representing the highest reported piezoelectric response on a stretchable medium. These results allow for the integration of energy conversion devices which operate in stretching mode via reversible deformations in the wavy/buckled ribbons.     

A piezoelectric screw dislocation interacting with a nonuniformly coated circular inclusion

This paper investigated the interaction of a piezoelectric screw dislocation with a nonuniformly coated circular inclusion in an unbounded piezoelectric matrix subjected to remote antiplane shear and electric fields. In addition to having a discontinuous displacement and a discontinuous electric potential across the slip plane, the dislocation is subjected to a line force and a line charge at the core. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general solutions in an explicit series form. This approach has a clear advantage in deriving the solution to the heterogeneous problem in terms of the solution for the corresponding homogeneous problem. The presented series solutions have rapid convergence which is guaranteed numerically. The image force acting on the piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. The numerical results show that the varying thickness of the interphase layer will exert a significant influence on the shear stress and electric field within the circular inclusion, and on the direction and magnitude of the image force. This solution can be used as Green’s function for the analysis of the corresponding piezoelectric matrix cracking problem.     

On the dynamic behavior of piezoelectric sensors and actuators embedded in elastic media

Summary Embedded piezoelectric sensors can be used to monitor the mechanical behaviour of structures for damage detection. This paper provides an analytical study of the dynamic behaviour of piezoelectric sensors embedded in elastic media under high frequency electromechanical loads induced by piezoelectric actuators. A generalized sensor/actuator model taking account of the deformation in both transverse and longitudinal directions of the piezoelectric sensor/actuator is developed. The dynamic load transfer between the sensors/actuators and the host medium is studied using Fourier transform method and solving the resulting integral equations in terms of the interfacial normal and shear stresses. Detailed numerical simulation is conducted to study the relation between the deformation of the sensor and that of the host medium under different loading conditions. The results show the significant effect of the geometry, the material combination and the loading frequency upon the behaviour of the sensor.