The effective electroelastic property of piezoelectric media with parallel dielectric cracks

Existing studies on the coupled electroelastic behaviour of cracked piezoelectric media have been based mostly on the electrically impermeable and permeable crack models. The current paper presents a study of the effective electroelastic property of piezoelectric media weakened by parallel cracks using a dielectric crack model with the electric boundary condition along the crack surfaces being governed by the opening displacement. The theoretical formulation is obtained using the dilute model of distributed cracks and the solution of a single dielectric crack problem. It is observed that the effective electroelastic property of cracked piezoelectric media is nonlinear and sensitive to loading conditions. Different modes of crack deformation are predicted and discussed. Attention is paid to the transition between electrically permeable and impermeable crack models.     

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K_I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.     

层状磁电复合材料界面共线裂纹平面问题分析

本文研究了面内电磁势载荷作用下双层压电压磁复合材料中共线界面裂纹问题．考虑了压电材料的导磁性质和压磁材料的介电性质,引入了界面电位移和磁感强度的连续性条件．利用Fourier 变换得到一组第二类Cauchy 型奇异积分方程．进一步导出了相应问题的应力强度因子、电位移强度因子和磁感强度强度因子的表达式,给出了应力强度因子的数值结果．结果表明电磁载荷会导致界面裂纹尖端I、II 混合型应力奇异性,同时还伴随着电位移和磁感强度的奇异性．比较了双裂纹左右端的应力强度因子,发现在面内极化方向上施加面内磁势载荷时共线裂纹内侧尖端区域的两个法向应力场发生互相干涉增强．     

Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media

A modified polarization saturation model is proposed and addressed mathematically using a complex variable approach in two-dimensional(2 D) semipermeable piezoelectric media. In this model, an existing polarization saturation(PS) model in 2D piezoelectric media is modified by considering a linearly varying saturated normal electric displacement load in place of a constant normal electric displacement load, applied on a saturated electric zone. A centre cracked infinite 2D piezoelectric domain subject to an arbitrary poling direction and in-plane electromechanical loadings is considered for the analytical and numerical studies. Here, the problem is mathematically modeled as a non-homogeneous Riemann-Hilbert problem in terms of unknown complex potential functions representing electric displacement and stress components. Having solved the Hilbert problem, the solutions to the saturated zone length, the crack opening displacement(COD), the crack opening potential(COP), and the local stress intensity factors(SIFs) are obtained in explicit forms. A numerical study is also presented for the proposed modified model, showing the effects of the saturation condition on the applied electrical loading, the saturation zone length, and the COP. The results of fracture parameters obtained from the proposed model are compared with the existing PS model subject to electrical loading, crack face conditions, and polarization angles.     

Dielectric Breakdown Model for an Electrically Semi-Permeable Penny-Shaped Crack in Three-Dimensional Piezoelectric Media

The dielectric breakdown(DB) model for a penny-shaped crack under a semipermeable boundary condition in a three-dimensional piezoelectric medium is studied.An approximate analytical solution is derived by using the boundary integral equation with extended displacement discontinuity,and the corresponding boundary element method with double iterative approaches is developed to analyze the semi-permeable crack.The effect of electric boundary conditions on crack faces is discussed on the basis of DB model.By comparing the DB model with the polarization saturation(PS) model for different piezoelectric materials,some interesting phenomena related to the electric yielding zone and local J-integral are observed.     

On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound

An interface crack between two semi-infinite piezoelectric spaces under the action of remote mixed mode loading and electric flux is considered. The properties of the materials, loading and crack geometry admit to consider a two-dimensional problem in the plane perpendicular to the crack front. The crack is assumed to be free from mechanical loading and the limited permeable electric condition holds true. Assuming the electric flux is constant along the crack area, using the known presentations of all electromechanical fields via a piecewise holomorphic vector function, the problem is reduced to a vector Hilbert problem and solved in an analytical way. Clear analytical expressions for stresses and electric displacement as well as for stress and electric intensity factors are derived. As a particular case, a crack in a homogeneous piezoelectric material is considered and exact analytical formulae are presented for this case. The numerical analysis of the obtained formulae showed that for small values of the electric flux the model of a completely permeable crack can be used for any real crack permeability’s. The validity of such an approximation decreases with increase in the mechanical loading and especially of the electric flux.     

Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer

The assumptions of impermeable and permeable cracks give rise to significant errors in determining electro-elastic behavior of a cracked piezoelectric material. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an opening crack interior. Considering the presence of the dielectric of an opening crack interior and the permeability of the crack surfaces for electric field, this paper analyzes electro-elastic behavior induced by a penny-shaped dielectric crack in a piezoelectric ceramic layer. In the cases of prescribed displacement or prescribed stress at the layer surfaces, the Hankel transform technique is employed to reduce the problem to Fredholm integral equations with a parameter dependent nonlinearly on the unknown functions. For an infinite piezoelectric space, a closed-form solution can be derived explicitly, while for a piezoelectric layer, an iterative technique is suggested to solve the resulting nonlinear equations. Field intensity factors are obtained in terms of the solution of the equations. Numerical results of the crack opening displacement intensity factors are presented for a cracked PZT-5H layer and the effect of applied electric field on crack growth are examined for both cases. The results indicate that the fracture toughness of a piezoelectric ceramic is affected by the direction of applied electric fields, dependent on the elastic boundary conditions.     

Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties

The paper presents a fracture analysis for an electromagnetically dielectric crack in a functionally graded magnetoelectroelastic strip. It is considered that the material properties are varying exponentially along the width direction. Under the assumption of the in-plane magneto-electro-mechanical loadings, the dielectric crack is simulated by using the semi-permeable crack-face boundary conditions. The Fourier transform technique is applied to solve the boundary-value problem and four coupling singular integral equations are determined. A nonlinear system of algebraic equations is further derived and solved numerically to determine the electromagnetic field inside the crack. Then the field intensity factors of stress, electric displacement, and magnetic induction are given. Through the numerical computations, the effects of the material non-homogeneity and the permeability of crack interior on the electric displacement and the magnetic induction at the crack faces are studied. The variations of the intensity factors of stress, electric displacement, and magnetic induction versus the geometry of the crack, the strip width, and the material non-homogeneity are presented in graphics respectively.     

General treatment of mode III interfacial crack problems in piezoelectric materials

Summary  The anti-plane problem of N collinear interfacial cracks between dissimilar transversely isotropic piezoelectric media, which are subjected to piecewise uniform out-of-plane mechanical loading combined with in-plane electric loading at infinity, and also a line loading at an arbitrary point, is addressed by using the complex function method. In comparison with other relevant works, the present study has two features: one is that the analysis is based on the permeable crack model, i.e. the cracks are considered as permeable thin slits, and, thus, both the normal component of electric displacement and the tangential component of electric field are assumed to be continuous across these slits. The other feature is that explicit closed-form solutions are given not only in piezoelectric media, but also inside cracks when the media are subjected to the most general loading. It is shown that the singularities of electric displacement and electric field in the media are always dependent on that of stress for the general case of loading, and all the singularities of field variables are independent of the applied uniform electric loads at infinity. For the interfacial cracks the electric field is square-root singular at the crack tips and shows jumps across the interface, while the normal component of the electric field is linearly variable inside the crack, but the tangential component is square-root singular. However, for a homogeneous medium with collinear cracks, the electric field is always nonsingular in the medium while the electric displacement exhibits square-root singularity. Moreover, in this case, the electric field inside any crack is equal to a constant when uniform loads are applied at infinity. Received 22 November 1999; accepted for publication 20 July 2000     

A dielectric crack in a functionally graded piezoelectric layer

Considering the dielectric effects inside a crack, the problem of an electrically dielectric crack in a functionally graded piezoelectric layer is addressed in this paper. The energetically consistent crack-face boundary conditions are utilized to analyze the effects of a dielectric of crack interior. Applying the Fourier transform technique, the boundary-value problem is reduced to solving three coupling singular equations. Then a system of non-linear algebraic equations is obtained and the field intensity factors along with the energy release rate are given. Numerical results show the differences of the electric displacement inside a crack, the stress and electric displacement intensity factors and the energy release rate using the permeable, impermeable, semi-permeable and energetically consistent boundary conditions respectively. The effects of the material non-homogeneity, the applied electric field and the discharge field of crack interior on the electrostatic traction acting on the crack faces and the energy release rate are further studied through the energetically consistent boundary conditions.     

Re-examination of the PKHS crack model in piezoelectric materials

In this paper, the PKHS crack model is re-examined in detail. Some novel features and better understandings are presented. It is found that the normal crack opening displacement jump (NCODJ) across the crack is always extremely small, whereas the electric potential drop (EPD) across the crack is very large for PZT-4 or PZT-5H under 20 MPa and a varying electric field from −0.5 MV/m to 0.5 MV/m. It is also found that the permittivities of the medium inside the crack gap influence the NCODJ and the EPD significantly. Of great significance is that the crack tip energy release rate (ERR) does not always show an even functional dependence on the applied electric field as previous researchers reported. Its variable tendencies (even or odd) depend on the mechanical loading levels, which are called as the load dependence feature here. In PZT-4, this load-dependence feature induced from the increasing mechanical loading may lead to a transformation of the crack tip ERR from an even functional dependence to an odd one when the applied electric loading varies within the range mentioned above, whereas in PZT-5H no such feature exists. This implies that different mechanical loading levels may yield different fracture behaviors when using the crack tip ERR as a fracture criterion. Moreover, the permittivities of medium inside the PKHS crack gap influence the variable tendencies of the ERR significantly. Thus, a useful addition to previous works is provided and a reasonable explanation for the fundamental discrepancy between previous theoretical predictions and experimental observations in piezoelectric fracture can be given. That is, previous experiments were performed for quite different piezoelectric materials, under quite different mechanical loading levels, and with quite different media inside crack gaps.     

Dynamic analysis of a penny-shaped dielectric crack in a magnetoelectroelastic solid under impacts

The transient response of a magneto-electro-elastic material with a penny-shaped dielectric crack subjected to in-plane magneto-electro-mechanical impacts is made. To simulate an opening crack with a dielectric interior, the crack-face electromagnetic boundary conditions are supposed to depend on the crack opening displacement and the jumps of electric and magnetic potentials across the crack. Four ideal crack-face electromagnetic boundary conditions involving a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions can be reduced. The Laplace and Hankel transform techniques are further utilized to solve the mixed initial-boundary-value problem. Three coupling Fredholm integral equations are obtained and solved by the composite Simpson's rule. Dynamic field intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD), electric potential and magnetic potential are given in the Laplace transform domain. By means of a numerical inversion of the Laplace transform, numerical results are calculated to show the variations of the physical parameters of concern versus the normalized time in graphics. The effects of applied electric and magnetic loads on the dynamic intensity factors of stress and COD, and the dynamic energy release rate for a BaTiO₃-CoFe₂O₄ composite with a penny-shaped vacuum crack are discussed in detail.     

Transient response of a magnetoelectroelastic solid with two collinear dielectric cracks under impacts

Dynamic analysis of two collinear electro-magnetically dielectric cracks in a piezoelectromagnetic material is made under in-plane magneto-electro-mechanical impacts. Generalized semi-permeable crack-face boundary conditions are proposed to simulate realistic opening cracks with dielectric. Ideal boundary conditions of a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions are several limiting cases of the semi-permeable dielectric crack. Utilizing the Laplace and Fourier transforms, the mixed initial-boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. Dynamic intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) near the inner and outer crack tips are determined in the Laplace transform domain. Numerical results for a special magnetoelectroelastic solid are calculated to show the influences of the dielectric permittivity and magnetic permeability inside the cracks on the crack-face electric displacement and magnetic induction. By means of a numerical inversion of the Laplace transform, the variations of the normalized intensity factors of stress and COD are discussed against applied magnetoelectric impact loadings and the geometry of the cracks for fully impermeable, vacuum, fully permeable cracks and shown in graphics.     

Periodic set of limited electrically permeable interface cracks with contact zones

Plane problem for an infinite space composed of two different piezoelectric or piezoelectric/dielectric semi-infinite spaces with a periodic set of limited electrically permeable interface cracks is considered. Uniformly distributed electromechanical loading is applied at infinity. The frictionless contact zones at the crack tips are taken into account. The problem is reduced to the combined Dirichlet–Riemann boundary value problem by means of the electromechanical factors presentation via sectionally analytic functions, assuming that the electric flux is uniformly distributed inside the cracks. An exact solution of the problem is proposed. It permits to find in a closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux value. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region.Formulae for stresses, electric displacement vector, elastic displacements and electric potential jump at the interface as well as the intensity factors at the crack tips are given. Equation for the contact zone length determination is presented. Calculations for certain material combinations are carried out. The influence of electric permeability of cracks on electromechanical fields and the fracture mechanical parameters is analyzed.     

Numerical solution of polarization saturation/dielectric breakdown model in 2D finite piezoelectric media

The polarization saturation (PS) model [Gao, H., Barnett, D.M., 1996. An invariance property of local energy release rates in a strip saturation model of piezoelectric fracture. Int. J. Fract. 79, R25–R29; Gao, H., Zhang, T.Y., Tong, P., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491–510], and the dielectric breakdown (DB) model [Zhang, T.Y., Zhao, M.H., Cao, C.F., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311–327] explain very well some experimental observations of fracture of piezoelectric ceramics. In this paper, the nonlinear hybrid extended displacement discontinuity-fundamental solution method (NLHEDD-FSM) is presented for numerical analysis of both the PS and DB models of two-dimensional (2D) finite piezoelectric media under impermeable and semi-permeable electric boundary conditions. In this NLHEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and the electric yielding zone. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy certain conditions on the boundary of the domain, on the crack face and the electric yielding zone. The zero electric displacement intensity factor in the PS model or the zero electric field strength intensity factor in the DB model at the outer tips of the electric yielding zone is used as a supplementary condition to determine the size of the electric yielding zone. Iteration approaches are adopted in the NLHEDD-FSM. The electric yielding zone is determined, and the extended intensity factors and the local J-integral are calculated for center cracks in piezoelectric strips. The effects of finite domain size, saturation property and different electric boundary conditions, as well as different models on the electric yielding zone and the local J-integral, are studied.     

Investigation of the crack problem in non-local piezoelectric materials under combined electromechanical loadings

The present investigation of the crack problem in piezoelectric materials is performed based on the non-local theory. After some manipulations, the impermeable crack, the permeable crack （the crack gap is full of NaCI solution）, and the semi-permeable crack （the crack gap is full of air or silicon oil） are reduced to a uniform formulation by assuming the normal electric displacement on the crack surfaces to be an unknown variable. Thus, a triple integral equation with the unknown normal electric displacement is established. By using the Newton iterative method and solving the triple integral equation, it is found that the normal electric displacement on the crack surfaces is no longer a constant as determined by previous studies, rather, it depends upon the remote combined electromechanical loadings. Numerical results of the stresses and electric displacement fields show that there are no singularities at the crack tips so that the stresses remain finite. It is of great significance that the concrete electric boundary condition on the crack surfaces exerts significant influence on the near-tip fields and in this way plays an important role in evaluating the crack stability in the non-local piezoelectric materials. More specifically, the impermeable crack model always overestimates the finite stresses at the crack tips, whereas the permeable crack model always underestimates them.     

Mode-III interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials

Summary  The problem of an interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials is considered under the conditions of anti-plane shear and in-plane electric loading. The crack surfaces are assumed to be impermeable to the electric field. An integral transform technique is employed to reduce the problem under consideration to dual integral equations. By solving the resulting dual integral equations, the intensity factors of the stress and the electric displacement and the energy release rate as well as the crack sliding displacement and the electric voltage across the crack surfaces are obtained in explicit form for the case of concentrated forces and free charges at the crack surfaces and at the boundary. The derived results can be taken as fundamental solutions which can be superposed to model more realistic problems. Received 10 November 2000; accepted for publication 28 March 2001     

Nonlinear fracture of 2D magnetoelectroelastic media: Analytical and numerical solutions

A strip electric–magnetic polarization saturation (SEMPS) model is developed to study the electric and magnetic yielding effects on a crack in magnetoelectroelastic (MEE) media. In this model, the MEE medium is treated as being mechanically brittle, and electrically and magnetically ductile. Analogously to the classic Dugdale model, the electric and magnetic yielding zones in front of the crack are represented for simplicity by two strips. In the electric yielding strip the electric displacement equals the electric displacement saturation and meanwhile in the magnetic yielding zone the magnetic induction equals the magnetic induction saturation. The nonlinear analytical solution of this SEMPS model of crack in an infinite MEE medium is obtained using an integral equation approach. The equivalence between the proposed SEMPS model and the existing strip electric–magnetic breakdown (SEMB) model is demonstrated.To analyze the nonlinear fracture problem in the corresponding finite MEE media, the non-linear hybrid extended displacement discontinuity-fundamental solution (NLHEDD-FS) method is modified, and a multiple iteration approach is adapted to determine the electric and magnetic yielding zones. Comparing with the analytical solution, the applicability and effectiveness of the NLHEDD-FS method is verified. Numerical results based on the SEMPS model for a center crack in infinite and finite MEE strip are presented.     

Analysis of electric boundary condition effects on crack propagation in piezoelectric ceramics

There are three types of cracks: impermeable crack, permeable crack and conducting crack, with different electric boundary conditions on faces of cracks in piezoelectric ceramics, which poses difficulties in the analysis of piezoelectric fracture problems. In this paper, in contrast to our previous FEM formulation, the numerical analysis is based on the used of exact electric boundary conditions at the crack faces, thus the common assumption of electric impermeability in the FEM analysis is avoided. The crack behavior and elasto-electric fields near a crack tip in a PZT-5 piezoelectric ceramic under mechanical, electrical and coupled mechanical-electrical loads with different electric boundary conditions on crack faces are investigated. It is found that the dielectric medium between the crack faces will reduce the singularity of stress and electric displacement. Furthermore, when the permittivity of the dielectric medium in the crack gap is of the same order as that of the piezoelectric ceramic, the crack becomes a conducting crack, the applied electric field has no effect on the crack propagation. The project supported by the National Natural Science Foundation of China (19672026, 19891180)     

THE BEHAVIOR OF TWO PARALLEL SYMMETRIC PERMEABLE CRACKS IN PIEZOELECTRIC MATERIALS

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