Electro-elastic Fields of Piezoelectric Materials with An Elliptic Hole under Uniform Internal Shearing Forces

The existing investigations on piezoelectric materials containing an elliptic hole mainly focus on remote uniform tensile loads. In order to have a better understanding of the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole in transversely isotropic piezoelectric materials subjected to uniform internal shearing forces based on the complex potential approach. By solving ten variable linear equations, the analytical solutions inside and outside the hole satisfying the permeable electric boundary conditions are obtained. Taking PZT-4 ceramic into consideration, numerical results of electro-elastic fields along the edge of the hole and axes, and the electric displacements in the hole are presented. Comparison with stresses in transverse isotropic elastic materials shows that the hoop stress at the ends of major axis in two kinds of material equals zero for the various ratios of major to minor axis lengths; If the ratio is greater than 1, the hoop stress in piezoelectric materials is smaller than that in elastic materials, and if the ratio is smaller than 1, the hoop stress in piezoelectric materials is greater than that in elastic materials; When it is a circle hole, the shearing stress in two materials along axes is the same. The distribution of electric displacement components shows that the vertical electric displacement in the hole and along axes in the material is always zero though under the permeable electric boundary condition; The horizontal and vertical electric displacement components along the edge of the hole are symmetrical and antisymmetrical about horizontal axis, respectively. The stress and electric displacement distribution tends to zero at distances far from the elliptical hole, which conforms to the conclusion usually made on the basis of Saint-Venant’s principle. Unlike the existing work, uniform shearing forces acting on the edge of the hole, and the distribution of electro-elastic fields inside and outside the elliptic hole are considered.

THE BEHAVIOR OF TWO PARALLEL SYMMETRIC PERMEABLE CRACKS IN PIEZOELECTRIC MATERIALS

The behavior of two parallel symmetric cracks in piezoelectric materials under anti-plane shear loading was studied by the Schmidt method for the permeable crack face conditions. By using the Fourier transform, the problem can be solved with two pairs of dual integral equations in which the unknown variable is the jump of the diplacement across the crack surfaces. These equations were solved using the Schmidt method. The results show that the stress and the electric displacement intensity factors of cracks depend on the geometry of the crack. Contrary to the impermeable crack surface condition solution, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.

THE PERIODIC CRACK PROBLEM IN BONDED PIEZOELECTRIC MATERIALS

The problem of a periodic array of parallel cracks in a homogeneous piezoelectric strip bonded to a functionally graded piezoelectric material is investigated for inhomogeneous continuum. It is assumed that the material inhomogeneity is represented as the spatial variation of the shear modulus in the form of an exponential function along the direction of cracks. The mixed boundary value problem is reduced to a singular integral equation by applying the Fourier transform, and the singular integral equation is solved numerically by using the Gauss-Chebyshev integration technique. Numerical results are obtained to illustrate the variations of the stress intensity factors as a function of the crack periodicity for different values of the material inhomogeneity.

ANALYSIS OF THE DYNAMIC BEHAVIOR OF A GRIFFITH PERMEABLE CRACK IN PIEZOELECTRIC MATERIALS WITH THE NON-LOCAL THEORY

The dynamic behavior of a Griffith permeable crack under harmonic anti-plane shear waves in the piezoelectric materials is investigated by use of the non-local theory. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By means of Fourier transform, the problem can be solved with a pair of dual integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved with the Schmidt method and numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularities are present at the crack tip. The finite hoop stress and the electric displacement depend on the crack length, the lattice parameter of the materials and the circle frequency of the incident waves. This enables us to employ the maximum stress hypothesis to deal with fracture problems in a natural way.

PENNY-SHAPED CRACK IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC MATERIALS

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the dis- placement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordi- nate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r^(-1/2)) singularity.

ANTI-PLANE SHEAR CRACK IN A FUNCTIONALLY GRADIENT PIEZOELECTRIC MATERIAL

The main objective of this paper is to study the singular natureof the crack-tip stress and electric displacement field in afunctionally gradient piezoelectric medium having materialcoefficients with a discontinuous derivative. The problem isconsidered for the simplest possible loading and geometry, namely,the anti-plane shear stress and electric displacement in -plane oftwo bonded half spaces in which the crack is parallel to theinterface.

Exact Solutions for Piezoelectric Materials with an Elliptic Hole or a Crack under Uniform Internal Pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads.In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions,theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads.On the basis of the complex variable approach,analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole.As an example of PZT-4 ceramics,numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given,and graphs of the electro-elastic fields in the vicinity of the crack tip are presented.The non-singular term is compared to the asymptotic one in the figures.It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack,and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity,but not on the electric loads at infinity.The figures obtained are strikingly similar to the available results.Unlike the existing work,the existence of electric fields inside an elliptic hole or a crack is considered,and the piezoelectric solid is subjected to complicated electro-mechanical loads.

AN EXACT SOLUTION OF CRACK PROBLEMS IN PIEZOELECTRIC MATERIALS

An assumption that the normal component of the electric displacement on crack faces is thought of as being zero is widely used in analyzing the fracture mechanics of piezoelectric materials. However, it is shown from the available experiments that the above assumption will lead to erroneous results. In this paper, the two-dimensional problem of a piezoelectric material with a crack is studied based on the exact electric boundary condition on the crack faces. Stroh formalism is used to obtain the closed-form solutions when the material is subjected to uniform loads at infinity. It is shown from these solutions that: (i) the stress intensify factor is the same as that of isotropic material, while the intensity factor of the electric displacement depends on both material properties and the mechanical loads, but not on the electric load. (ii) the energy release rate in a piezoelectric material is larger than that in a pure elastic-anisotropic material, i.e., it is always positive, and independent of the electric loads. (iii) the field solutions in a piezoelectric material are not related to the dielectric constant of air or vacuum inside the crack.

DYNAMIC BEHAVIOR OF TWO UNEQUAL PARALLEL PERMEABLE INTERFACE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF PIEZOELECTRIC MATERIALS PLANES

The dynamic behavior of two unequal parallel permeable interface cracks in a piezoelectric layer bonded to two half-piezoelectric material planes subjected to harmonic anti-plane shear waves is investigated. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables were the jumps of the displacements across the crack surfaces. Numerical results are presented graphically to show the effects of the geometric parameters, the frequency of the incident wave on the dynamic stress intensity factors and the electric displacement intensity factors. Especially, the present problem can be returned to static problem of two parallel permeable interface cracks. Compared with the solutions of impermeable crack surface condition, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller.

Two-dimensional fracture analysis of piezoelectric material based on the scaled boundary node method

A scaled boundary node method （SBNM） is developed for two-dimensional fracture analysis of piezoelectric material, which allows the stress and electric displacement intensity factors to be calculated directly and accurately. As a boundary- type meshless method, the SBNM employs the moving Kriging （MK） interpolation technique to an approximate unknown field in the circumferential direction and therefore only a set of scattered nodes are required to discretize the boundary. As the shape functions satisfy Kronecker delta property, no special techniques are required to impose the essential boundary conditions. In the radial direction, the SBNM seeks analytical solutions by making use of analytical techniques available to solve ordinary differential equations. Numerical examples are investigated and satisfactory solutions are obtained, which validates the accuracy and simplicity of the proposed approach.

Piezoelectric Materials for Controlling Electro-Chemical Processes

Piezoelectric materials have been analyzed for over 100 years,due to their ability to convert mechanical vibrations into electric charge or electric fields into a mechanical strain for sensor,energy harvesting,and actuator applications.A more recent development is the coupling of piezoelectricity and electro-chemistry,termed piezo-electro-chemistry,whereby the piezoelectrically induced electric charge or voltage under a mechanical stress can influence electro-chemical reactions.There is growing interest in such coupled systems,with a corresponding growth in the number of associated publications and patents.This review focuses on recent development of the piezo-electro-chemical coupling multiple systems based on various piezoelectric materials.It provides an overview of the basic characteristics of piezoelectric materials and comparison of operating conditions and their overall electro-chemical performance.The reported piezo-electro-chemical mechanisms are examined in detail.Comparisons are made between the ranges of material morphologies employed,and typical operating conditions are discussed.In addition,potential future directions and applications for the development of piezo-electro-chemical hybrid systems are described.This review provides a comprehensive overview of recent studies on how piezoelectric materials and devices have been applied to control electro-chemical processes,with an aim to inspire and direct future efforts in this emerging research field.

SCATTERING OF THE HARMONIC STRESS WAVE BY CRACKS IN FUNCTIONALLY GRADED PIEZOELECTRIC MATERIALS

The present paper considers the scattering of the time harmonic stress wave by a single crack and two collinear cracks in functionally graded piezoelectric material （FGPM）. It is assumed that the properties of the FGPM vary continuously as an exponential function. By using the Fourier transform and defining the jumps of displacements and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the influences of material properties on the dynamic stress and the electric displacement intensity factors.

INVESTIGATION OF BEHAVIOR OF MODE-I INTERFACE CRACK IN PIEZOELECTRIC MATERIALS BY USING SCHMIDT METHOD

The behavior of a Mode-Ⅰ interface crack in piezoelectric materials was investigated under the assumptions that the effect of the crack surface overlapping very near the crack tips was negligible. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. To solve the dual integral equations, the jumps of the displacements across the crack surfaces were expanded in a series of Jacobi polynomials. It is found that the stress and the electric displacement singularities of the present interface crack solution are the same as ones of the ordinary crack in homogenous materials. The solution of the present paper can be returned to the exact solution when the upper half plane material is the same as the lower half plane material.

AN INTERFACE INCLUSION BETWEEN TWO DISSIMILAR PIEZOELECTRIC MATERIALS

The generalized two-dimensional problem of a dielectric rigid line inclusion, at the interface between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity, is studied by means of the Stroh theory. The problem was reduced to a Hilbert problem, and then closed-form expressions were obtained, respectively, far the complex potentials in piezoelectric media, the electric field inside the inclusion and the tip fields near the inclusion. it is shown that in the media, all field variables near the inclusion-tip show square root singularity and oscillatory singularity, the intensity of which is dependent on the material constants and the strains at infinity. In addition, it is found that the electric field inside the inclusion is singular and oscillatory too, when approaching the inclusion-tips from inside the inclusion.

ELECTRO-ELASTIC FUNDAMENTAL SOLUTIONS OF ANISOTROPIC PIEZOELECTRIC MATERIALS WITH AN ELLIPTICAL HOLE

By using Stroh' complex formalism and Cauchy's integral method, the electro-elastic fundamental solutions of an infinite anisotropic piezoelectric solid containing an elliptic hole or a crack subjected to a Line force and a line charge are presented in closed form. Particular attention is paid to analyzing the characteristics of the stress and electric displacement intensity factors. When a line force-charge acts on the crack surface, the real form expression of intensity factors is obtained. It is shown through a special example that the present work is correct.

Dual equations and solutions of I-type crack of dynamic problems in piezoelectric materials

This paper firstly works out basic differential equations of piezoelectric materials expressed in terms of potential functions, which are introduced in the very beginning. These equations are primarily solved through Laplace transformation, semiinfinite Fourier sine transformation and cosine transformation. Secondly, dual equations of dynamic cracks problem in 2D piezoelectric materials are established with the help of Fourier reverse transformation and the introduction of boundary conditions. Finally, according to the character of the Bessel function and by making full use of the Abel integral equation and its reverse transform, the dual equations are changed into the second type of Fredholm integral equations. The investigation indicates that the study approach taken is feasible and has potential to be an effective method to do research on issues of this kind.

ANALYSIS OF A PARTIALLY DEBONDED ELLIPTIC INHOMOGENEITY IN PIEZOELECTRIC MATERIALS

A generalized solution was obtained for the partially debonded elliptic inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric loading using the complex variable method. It was assumed that the interfacial debonding induced an electrically impermeable crack at the interface. The principle of conformal transformation and analytical continuation were employed to reduce the formulation into two Riemann-Hilbert problems. This enabled the determination of the complex potentials in the inhomogeneity and the matrix by means of series of expressions. The resulting solution was then used to obtain the electroelastic fields and the energy release rate involving the debonding at the inhomogeneity-matrix interface. The validity and versatility of the current general solution have been demonstrated through some specific examples such as the problems of perfectly bonded elliptic inhomogeneity, totally debonded elliptic inhomogeneity, partially debonded rigid and conducting elliptic inhomogeneity, and partially debonded circular inhomogeneity.

Basic solution of two parallel non-symmetric permeable cracks in piezoelectric materials

The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading was studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations in which the unknown variables are the jumps of displacements across crack surfaces. To Solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.

Improved reproducing kernel particle method for piezoelectric materials

In this paper, the normal derivative of the radial basis function （RBF） is introduced into the reproducing kernel particle method （RKPM）, and the improved reproducing kernel particle method （IRKPM） is proposed. The method can decrease the errors on the boundary and improve the accuracy and stability of the algorithm. The proposed method is applied to the numerical simulation of piezoelectric materials and the corresponding governing equations are derived. The numerical results show that the IRKPM is more stable and accurate than the RKPM.

ANTI-PLANE CRACK MOVING ALONG THE INTERFACE OF DISSIMILAR PIEZOELECTRIC MATERIALS

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement around the crack tip are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will be reduced to the result far the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials.