Antiplane Permeable Edge Cracks in a Piezoelectric Strip of Finite Width

A piezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

Fatigue crack propagation in a piezoelectric ceramic strip subjected to mode III loading

Summary Following the theory of linear piezoelectricity, a forth-power stress intensity factor crack growth equation in an orthotropic piezoelectric ceramic strip is developed under mode III loading. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Dugdale's assumption regarding the plastic zone in metals is applied to estimate the effects of yield around the crack tips. Fourier transforms are used to reduce the electroelastic problem to one involving the numerical solution of a Fredholm integral equation of the second kind. A direct approach based on the accumulated plastic displacement criterion for crack propagation is used to develop the equation to predict the fatigue crack growth. Graphical results showing the effect of electroelastic interactions on the fatigue crack growth rate are presented.     

Antiplane permeable edge cracks in a piezoelectric strip of finite width

A priezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

Transient response of a functionally graded piezoelectric strip with a penny-shaped crack under electric time-dependent loading

Summary The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a penny-shaped crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under time-dependent electric load. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location and the material nonhomogeneity.     

Two parallel penny-shaped or annular cracks in a functionally graded piezoelectric strip under electric loading

In this paper, the mixed-mode fracture problem of a functionally graded piezoelectric material strip with two penny-shaped or annular cracks is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under electric loading. The problem is formulated in terms of a system of singular integral equations, which are solved numerically. Numerical calculations are carried out, and the stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

Electroelastic properties of cracked piezoelectric materials under longitudinal shear

Analytical solutions are obtained to quantify the influence of cracks on electroelastic properties of piezoelectric materials containing doubly-periodic arrays of cracks. Both the rectangular and diamond-shaped arrays of cracks are considered. Solutions are obtained for the case of an antiplane shear load coupled with an in-plane electrical load. This study makes it possible to understand the multicrack interactions in piezoelectric solids and their effects on the fracture and electroelastic properties. The crack tip field intensity factors and the change in stored electroelastic energy due to the presence of many microcracks are calculated. These calculations enable the prediction of the effective elastic, piezoelectric and dielectric constants of a damaged piezoelectric material. The results of this work can be useful in developing a technique to determine the state of mechanical and electrical damage in piezoelectric materials.     

A Moving Mode-III Crack in a Functionally Graded Piezoelectric Strip

In this paper, the problem of a functionally graded piezoelectric strip with a constant-velocity Yoffe-type moving crack is considered. By using the Fourier transforms, the problem is first reduced to dual integral equations and then into Fredholm integral equations of the second kind. The electroelastic field near the crack tip is obtained for electrical impermeable boundary conditions and electrical permeable boundary conditions, respectively. The results obtained show that the gradient of the material properties can increase or decrease the magnitudes of the stress intensity factors, and the velocity can disturb the stress distribution near the crack tip.     

On the dynamic propagation of an anti-plane shear crack in a functionally graded piezoelectric strip

Summary. In this paper, a finite crack propagating at constant speed in a functionally graded piezoelectric material (FGPM) is studied. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential gradients along the thickness of the strip, and that the strip is under anti-plane shear mechanical and in-plane electrical loads. The analysis is conducted on the electrically unified (natural) crack boundary condition, which is related to the ellipsoidal crack parameters. By using the Fourier transform, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, crack propagation speed, electric field, FGPM gradation, crack length, and electromechanical coupling coefficient. It reveals that there are considerable differences between traditional electric crack models and the present unified crack model.     

Impact response of an anti-plane crack in a FGPM bonded to a homogeneous piezoelectric strip

A finite crack under transient anti-plane shear loads in a functionally graded piezoelectric material (FGPM) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential functions along the thickness of the strip, and that the two layered strips is under combined anti-plane shear mechanical and in-plane electrical impact loads. The analysis is conducted on the electrically unified crack boundary condition. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Fredholm integral equations of the second kind in the Laplace transform domain. Then, a numerical Laplace inversion is performed and the dynamic intensities are obtained as functions of time and geometric parameters, which are displayed graphically.     

Transient response of a center crack in a functionally graded piezoelectric strip under electromechanical impact

The dynamic fracture problem for a functionally graded piezoelectric strip containing a center crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the medium vary continuously in the thickness direction, and that the strip is under in-plane mechanical and electric impact loadings. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The dynamic stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the material nonhomogeneity and the loading combination.     

Effects of remanent field on an elliptical flaw and a crack in a poled piezoelectric ceramic

Commonly used piezoelectric ceramics such as PZT and PLZT are polarized ferroelectric polycrystals. After poling, remanent strains and a remanent polarization exist in a ceramic material. Remanent field can affect the electroelastic field and consequently plays a critical role in fracture of poled ceramics. Based on a linear constitutive law, the electroelastic field and the energy release rate of an elliptical cavity (or a crack) in a poled piezoelectric are re-examined in this study by including the effects of remanent field. It is noted that the remanent field generally has a minor effect on the stress field and a pronounced effect on the electric field at the apex of the major axis of an elliptical flaw. When the permittivity of the cavity is small, the effect of remanent polarization is similar to that of a very strong electric field applied along the poling direction. However, for the case of a conducting flaw, the remanent field does not influence the electroelastic field and energy release rate. Energy release rate of a flaw in a poled ferroelectric ceramic with and without the remanent polarization is generally different.     

The effective electroelastic property of cracked piezoelectric media

Summary This paper presents a study on the effective electroelastic property of piezoelectric media with parallel or randomly distributed cracks. The theoretical formulation is derived using the dilute model of distributed cracks and the solution of a single dielectric crack problem, in which the electric boundary condition along the crack surfaces is governed by the crack opening displacement. It is observed that the effective electroelastic property of such cracked piezoelectric media is nonlinear and sensitive to loading conditions. Numerical simulations are conducted to show the effects of crack distribution and electric boundary condition upon the effective electroelastic property. The transition between the commonly used electrically permeable and impermeable crack models is studied.     

The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading

The primary objective of this paper is to study the influence of the electroelastic interactions on the stress intensity factor in bonded layers of piezoelectric and orthotropic materials containing a crack along the interface under antiplane shear. Attention is given to a two-layer hybrid laminate formed by adding a layer of piezoelectric ceramic to a unidirectional graphite/epoxy composite or an aluminum layer. Electric displacement or electric field is prescribed on the surfaces of the piezoelectric layer. The problem is formulated in terms of a singular integral equation which is solved by using a relatively simple and efficient technique. A number of examples are given for various material combinations. The results show that the effect of the electroelastic interactions on the stress intensity factor and the energy release rate can be highly significant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Closed form solutions for partially debonded circular inclusion in piezoelectric materials

Summary This paper deals with the problem of a partially debonded piezoelectric circular inclusion in a piezoelectric matrix. This boundary value problem is reduced to two Riemann-Hilbert problems through the use of the analytical continuation theory.Closed form solutions are obtained by considering the behavior of the complex field potentials at origin and infinity. The formulae for the electro-elastic field intensity factors of the interfacial crack are derivedexplicitly. Several particular cases are provided to show the effect of the crack angle, the mechanical and electrical properties and the loads on the electroelastic field singularities.     

Cylindrical interface cracks in 1-3 piezocomposites

1-3 Piezocomposites are made by embedding piezoelectric fibers/rods in polymer matrix materials. Fiber–matrix interface fracture can affect the performance of piezocomposites. In this paper, axisymmetric interfacial cracks in piezocomposites are studied by considering an idealized model of a single piezoelectric fiber in a matrix material. The displacement discontinuity method is used to formulate the Mode I and II crack problems. The fundamental solutions required for DDM are derived explicitly by using the electroelastic field equations and Fourier integral transforms. The dependence of Mode I and II stress intensity factors of single and multiple interface cracks on fiber and matrix material properties, crack length and distance between cracks are investigated.     

Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads

The problem of a semi-infinite impermeable mode-III crack in a piezoelectric material is considered under the action of impact loads. For the case when a pair of concentrated anti-plane impact loads and electric displacements are exerted symmetrically on the upper and lower surfaces of the crack, the asymptotic electroelastic field ahead of the crack tip is determined in explicit form. The dynamic intensity factors of electroelastic field and dynamic mechanical strain energy release rate are obtained. The obtained results can be taken as fundamental solutions, from which general results may directly be evaluated by integration. The method adopted is to reduce the mixed initial-boundary value problem, by using the Laplace and Fourier transforms, into two simultaneous dual integral equations. One may be converted into an Abel's integral equation and the other into a singular integral equation with Cauchy kernel, and the solutions of both equations can be determined in closed-form, respectively. For some particular cases, the present results reduce to the previous results.     

The strip dielectric breakdown model

This paper reports on the analysis of the strip dielectric breakdown (DB) model for an electrically impermeable crack in a piezoelectric medium based on the general linear constitutive equations. The DB model assumes that the electric field in a strip ahead of the crack tip is equal to the dielectric breakdown strength, which is in analogy with the classical Dugdale model for plastic yielding. Using the Stroh formalism and the dislocation modeling of a crack, we derived the relationship between the DB strip size and applied mechanical and electrical loads, the intensity factors of stresses and electric displacement, and the local energy release rate. Based on the results, we discussed the effect of electric fields on fracture of a transversely isotropic piezoelectric ceramic by applying the local energy release rate as a failure criterion. It is shown that for an impermeable crack perpendicular to the poling direction, a positive electric field will assist an applied mechanical stress to propagate the crack, while a negative electric field will retard crack propagation. However, for an impermeable crack parallel to the poling direction, it is found that the applied electric field does not change the mode I stress intensity factor and the local energy release rate, i.e., the applied electric field has no effect on the crack growth.     

Crack path selection in piezoelectric bimaterials

The problem of crack path selection in piezoelectric bimaterials is considered in this paper. Based on the Stroh formulation for anisotropic material and Green’s functions for piezoelectric bimaterials, the crack problem is expressed in terms of coupled singular integral equations at first, and then the equations are used to solve for stress and electric displacement fields numerically. A crack impinging an interface joining two dissimilar materials may arrest or may advance by either penetrating the interface or deflecting into it. The competition between deflection and penetration is investigated using the maximum energy release rate criterion. Numerical results are presented to study the role of remote electroelastic loads on the path selection of crack extension.     

Electroelastic response of a flat annular crack in a piezoelectric fiber surrounded by an elastic medium

Following the theory of linear piezoelectricity, the electroelastic problem of a flat annular crack in a piezoelectric fiber embedded in an elastic medium is considered. Fourier and Hankel transform techniques are employed to formulate the mixed-boundary-value problem as a singular integral equation. The stress-intensity factor, energy-release rate and energy-density factor are computed for some piezoelectric composites, and the influence of applied electric fields on the normalized values is displayed graphically.     

Semi-Infinite Anti-Plane Crack in a Piezoelectric Material

The problem of an impermeable semi-infinite crack in a piezoelectric material is considered. The electroelastic field due to a pair of concentrate forces and free charges applied at the crack surfaces is obtained by using the Mellin transform method, and the intensity factors of the quantities of concern and the mechanical strain energy release rate are given. The obtained results may be used as the fundamental solution, from which some general solutions can be calculated by superposition. The well-known solution for a purely elastic material will be recovered from the present solution if letting the coupling constant vanish.