Perturbation method of the piezoelectric fracture mechanics considering the permittivity of the medium in the crack gap

In this paper, the problem of the cracks with arbitrary forms in piezoelectric material is studied. The permittivity of the medium in the crack gap is considered. Except the collinear cracks, this boundary condition is too difficult to deal with; therefore, a perturbation method is recommended. By the way, the electric boundary conditions of electric fracture mechanics are discussed. For example, a small parameter solution of a crack is given and compared with the known `exact' (it will be discussed later) solution. This result shows that the impermeable or permeable conditions are only the boundary conditions for the first approximations of the perturbation solutions.     

Periodical collinear air containing cracks in a piezoelectric material

This paper derives an exact solution of periodical collinear cracks in a piezoelectric material plane. Two cases have been studied. In the first case, the permittivity of air _ais far less than that of piezoelectric material _m. Therefore, the electric induction in the air (Pak, 1990; Suo et al., 1992) may be neglected. In the second case, the permittivity of air is comparable with that of piezoelectric material. This electric induction is considered. Using the conformal mapping method, this problem can be deduced into a Riemann–Hilbert problem and solved in closed form. It is shown that the exact linear theory results (McMeeking, 1989; Gun and Fan, 1999) can be accepted when the remote stress ₂and ratio _m/_aare small. If they are large, the approximate nonlinear boundary conditions (Deeg, 1980; Pak, 1990; Suo et al., 1992) are suitable.     

3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed–mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given.     

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.     

Electro-elastic behavior induced by an external circular crack in a piezoelectric material

The electro-elastic problem of a transversely isotropic piezoelectric material with a flat crack occupying the outside of a circle perpendicular to the poling axis is considered in this paper. By using the Hankel transform technique, a mixed boundary value problem associated with the considered problem is solved analytically. The results are presented in closed form both for impermeable crack and for permeable crack. A full field solution is given, i.e., explicit expressions for electro-elastic field at any point in the entire piezoelectric space, as well as field intensity factors near the crack front, are determined. A numerical example for a cracked PZT-5H ceramic is given, and the effects of applied electric fields on elastic and electric behaviors are presented graphically.     

Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium

In this paper, the influence of the electric boundary conditions on cracks in piezoelectric components shall be studied. Several electric boundary conditions have been proposed in the literature. Here, influence of the permeability of the crack on electric and mechanical fields near the crack tip is considered. Cracks of lower permeability lead to stronger electric singularities. Furthermore, the influence on the stress intensity factors and energy release rate will be discussed. Finally, an experiment with piezoceramic CT specimens, which was performed by Park and Sun, will be evaluated taking into account the permeability of the crack.     

On contact zone models for an electrically impermeable interface crack in a piezoelectric bimaterial

An electrically impermeable interface crack between two semi-infinite piezoelectric planes under remote mechanical tension-shear and electrical loading is studied. Assuming the stresses, strains and displacements are independent on the coordinate x ₂ the expressions for the elastic displacement and potential jumps as well as for the stresses and electrical displacement along the interface via a sectionally holomorphic vector function are found. Introducing an artificial contact zone at the right crack tip and assuming the materials possess the symmetry class 6 mm the problem is reduced for a wide range of bimaterial compounds to a combination of combined Dirichlet–Riemann and Hilbert boundary value problems which are solved analytically. From these solutions clear analytical expressions for characteristic mechanical and electrical parameters are derived. As particular cases of the above mentioned solution the classical (oscillating) and contact zone solutions are obtained. Further, a comparison with an associated solution for an electrically permeable crack has been performed. The fracture mechanical parameters for all models via the remote loads are found analytically and important relationships between these parameters are obtained. Due to these relationships an important algorithm of a numerical method applicable for the investigation of an interface crack in a finite sized piezoelectric bimaterial is suggested.     

An energetically consistent annular crack in a piezoelectric medium

The analytical treatment of an energetically consistent annular crack in a piezoelectric solid subjected to remote opening electromechanical loading is addressed. Potential functions and Hankel transform in combination with a robust technique are employed to reduce the solution of the mixed boundary value problem into a Fredholm integral equation of the second kind. The limiting case of a penny-shaped crack in a piezoelectric medium with energetically consistent boundary conditions over the crack faces is extracted for the first time. The electrical discharge phenomenon within the crack gap is modeled utilizing a non-linear constitutive law and the effects of the breakdown field on the energy release rate are delineated. The energy release rate, the electric displacement inside the crack gap, and the closing traction on crack faces are plotted for all possible geometries of a non-discharging annular crack.     

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.     

Investigation of anti-plane shear behavior of a Griffith crack in a piezoelectric material by using the non-local theory

In this paper, the behavior of a Griffith crack in a piezoelectric material under anti-plane shear loading is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with two pairs of dual integral equations. These equations are solved using Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present at the crack tip.     

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.     

Griffith crack moving along the interface of two 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 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 reduce to the result for the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mode I crack in a soft ferromagnetic material

ABSTRACT In the existing magnetoelastic theories, stress is proportional to the square of magnetic intensity and the linear model developed is usually used to analyse magnetoelastic problems. For a crack problem, the perturbation of the magnetic field caused by deformation is not much less than the applied field. In this paper, complex potentials for a mode I crack with a nonlinear relation for magnetic intensity are developed. The boundary conditions on crack faces are represented in terms of the continuity of the magnetic field. A solution of the crack problem is obtained by solving the Riemann‐Hilbert problem. Making use of the solution, the effects of the boundary conditions on the crack faces on the magnetoelastic coupling are discussed.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

Transient thermoelectroelastic response of a functionally graded piezoelectric strip with a penny-shaped crack

Transient response of a penny-shaped crack in a plate of a functionally graded piezoelectric material (FGPM) is studied under thermal shock loading conditions. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the crack faces are completely insulated. By using both the Laplace and Hankel transforms, the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations which are solved numerically. The intensity factors vs. time for various crack size, crack position and material nonhomogeneity are obtained.     

Transient analysis of dynamic crack propagation in piezoelectric materials

Abstract

In this paper, the transient analysis of semi‐infinite propagating cracks in piezoelectric materials subjected to dynamic anti‐plane concentrated body force is investigated. The crack surface is assumed to be covered with an infinitesimally thin, perfectly conducting electrode that is grounded. In analyzing this problem, it has characteristic lengths and a direct attempt towards solving this problem by transform and Wiener‐Hopf techniques (Noble, 1958) is not applicable. In order to solve this problem, a new fundamental solution for propagating cracks in piezoelectric materials is first established and the transient response of the propagating crack is obtained by superposition of the fundamental solution in the Laplace transform domain. The fundamental solution to be used is the responses of applying exponentially distributed traction in the Laplace transform domain on the propagating crack surface. Taking into account the quasi‐static approximation, exact analytical transient solutions for the dynamic stress intensity factor and the dynamic electric displacement intensity factor are obtained by using the Cagniard‐de Hoop method (Cagnard, 1939; de Hoop, 1960) of Laplace inversion and are expressed in explicit forms. Numerical calculations of dynamic intensity factors are evaluated and the results are discussed in detail. The transient solutions for stationary cracks have been shown to approach the corresponding static values after the shear wave of the piezoelectric material has passed the crack tip.     

压电复合材料中III型唇形裂纹问题的解析解

采用复变函数方法和保角映射技术,研究了压电复合材料中含唇形裂纹的无限大体远场受反平面机械载荷和面内电载荷作用下的反平面问题,利用复变函数中的留数定理和Cauchy积分公式,分别获得了电不可通和电可通两种边界条件下裂纹尖端场强度因子和机械应变能释放率的解析表达式。当唇形裂纹的高度趋于零时,可得到无限大压电复合材料中Griffith裂纹的解析解。若不考虑电场作用,所得解退化为经典材料的已知结果。数值算例显示了裂纹的几何尺寸和机电载荷对机械应变能释放率的影响规律。结果表明: 唇形裂纹高度的增加会阻碍裂纹的扩展;机械载荷总是促进裂纹的扩展;电载荷对裂纹扩展的影响与裂纹面电边界条件有关。     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Modeling of the electromechanical coupling between crack propagation and piezoelectric behavior in active layers of smart devices

In this article, a new computational approach is investigated to predict the crack propagation inside some smart structures equipped with surface-bonded piezoelectric layers; meanwhile, the electromechanical coupling is exploited. The current industrial need of analyzing rather irregular geometries motivates resorting to a numerical approach. Therefore, the finite element method (FEM) is used to predict both the fracture behavior and the coupled response of piezoelectric material through a suitable interoperation of the two computational environments, available in some commercial code like the ABAQUS^©. The two solutions are then coupled by means of a subroutine, in this case operated through the ISIGHT^© tool. After a preliminary analysis and a validation, results of some numerical simulations are shown to highlight some significant peculiarities of the coupled behavior of the piezoelectric material.