Strip electric-magnetic breakdown model in a magnetoelectroelastic medium

Extending the polarization saturation model [Gao et al., 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 et al., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311-327] in piezoelectric materials, the Strip Electric-Magnetic Breakdown (SEMB) model is proposed for electrically and magnetically impermeable crack in a magnetoelectroelastic medium to study the effect of the nonlinear character of electric field and magnetic field on fracture of magnetoelectroelastic materials. In the SEMB model, the electric field in the strip of the electric breakdown zone ahead of the crack tip is equal to the electric breakdown strength, while the magnetic filed in the strip of the magnetic breakdown zone is equal to the magnetic breakdown strength. By using the extended Stroh formalism and the extended dislocation modeling of a crack, the Griffith crack problem under the electrically and magnetically elastic-plastic condition in a magnetoelectroelastic medium is reduced to a set of dual integral equations. The sizes of the electric breakdown zone and the magnetic breakdown zone, the extended intensity factors and the local J-integral are obtained. The effect of the combined mechanical-electric-magnetic loadings on the local J-integral is studied.     

Exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material

Based on the complex variable method and the technique of conformal mapping, the anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material is studied. The exact solutions of field intensity factors and energy release rate are presented in closed-form with the assumption that the surfaces of the cracks and the elliptical hole are electrically impermeable. With the variation of the hole-size and the crack length, the present results can be reduced to the cases of two symmetrical edge cracks and a single edge crack emanating from a circular hole given by Wang and Gao [Wang, Y.J., Gao, C.F., 2008. The mode III cracks originating from the edge of a circular hole in a piezoelectric solid. International Journal of Solids and Structures 45, 4590–4599]. Moreover, new models used for simulating more practical defects in a piezoelectric solid are obtained, such as two symmetrical edge cracks and a single edge crack emanating from an elliptical hole, two asymmetrical edge cracks emanating from a circular hole, T-shaped crack, cross-shaped crack and semi-infinite plane with an edge crack. Numerical examples are then conducted to reveal the effects of the hole-size and the crack length on the field intensity factors and the energy release rate.     

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.     

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)     

压电材料裂纹顶端条状电饱和区模型的力学分析

在线性压电本构方程框架下,对裂纹顶端条状电饱和区模型进行了严格的数学分析．完整地考虑了各向异性力电耦合效应．建立了电饱和区尺寸与外加电场的依赖关系．证实了当裂纹垂直极化轴时,压电材料的断裂应力随着外加正电场的增加而减小,随着外加负电场的增加而增加．当裂纹平行于极化轴时,与极化轴平行的外加电场对断裂应力无影响     

APPLICABILITY OF THE CRACK FACE ELECTRICAL BOUNDARY CONDITIONS IN PIEZOELECTRIC MECHANICS

The electrical boundary conditions on the crack faces and their applicability in piezoelectric materials are discussed. A slit crack and a notch of ?nite thickness in piezoelectric materials subjected to combined mechanical and electrical loads is consi…     

Green’s functions for anti-plane deformations of a circular arc-crack at the interface of piezoelectric materials

Summary The anti-plane deformation problem of an interfacial debounding crack between a circular piezoelectric inclusion and a piezoelectric matrix is investigated by means of the complex variables method. For a line load applied within the matrix or inside the inclusion, Greens functions are presented for the complex potentials, intensity factors and electric fields on the crack faces, respectively, in closed and explicit form. The solutions are valid for both permeable and impermeable crack models. It is shown that, in the general case of permeable cracks, the electric field singularity is always proportional to the stress singularity.The first author (C.F.Gao) would like to express his gratitude for the support of the Alexander von Humboldt Foundation (Germany).     

压电材料中裂纹面电边界条件的适用性

先分析理想裂纹面电边界条件(包括导通型裂纹和不导通型裂纹),然后分析缺陷的厚度及其对尖端场的影响,用封闭形式的解表示出了缺陷厚度的影响．结果指出,裂纹长厚比对计算结果是有影响的,除非裂纹极扁,不可通裂纹边界条件可以给出合理的结果,而可通边界条件只有在一些限制条件下才适用。     

压电介质表面圆形电极边缘的电-弹场分析

基于线性压电理论，研究了表面附有圆形电极板的半无限压电介质电极边缘的电弹场奇异分布特性。利用Hankel变换技术，将空间轴对称混合边值问题简化为一组对偶积分方程的求解，并得到了应力分量和电位移分量的解析表达式。算例表明在电极边缘的邻域内存在剧烈的应力集中和电位移增强，足以造成介质的脆性断裂或介质击穿。并对这种类裂纹奇异性质进行了讨论。     

Isolated crack in three-dimensional piezoelectric solid. Part II: Stress intensity factors for circular crack

This is part II of the work concerned with finding the stress intensity factors for a circular crack in a solid with piezoelectric behavior. The method of solution involves reducing the problem to a system of hypersingular integral equations by application of the unit concentrated displacement discontinuity and the unit concentrated electric potential discontinuity derived in part I [1]. The near crack border elastic displacement, electric potential, stress and electric displacement are obtained. Stress and electric displacement intensity factors can be expressed in terms of the displacement and the potential discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrary shaped cracks in a piezoelectric and elastic medium such that once the stress intensity factors in the piezoelectric medium can be determined directly from that of the elastic medium. Results for the penny-shaped crack are obtained as an example.     

Comments on “Explicit transient solutions for a mode III crack subjected to dynamic concentrated loading in a piezoelectric material” by Yi-Shyong Ing, Mau-Jung Wang [Int. J. Solids Struct. 41 (2004) 3849–3864]

In this comment it is pointed out that the analysis of the dynamic stress intensity factor, dynamic electric displacement intensity factor and dynamic energy release rate conducted by Ing and Wang [Ing, Y.S., Wang, M.J., 2004. Explicit transient solutions for a mode III crack subjected to dynamic concentrated loading in a piezoelectric material. International Journal of Solids and Structures 41, 3849–3864] is incorrect. The correct analysis and corresponding correct plots are presented.     

压电材料裂纹顶端条状电饱和区模型的力学分析

在线性压电本构方程框架下,对裂纹顶端条状电饱和区模型进行了严格的数学分析．完整地考虑了各向异性力电耦合效应．建立了电饱和区尺寸与外加电场的依赖关系．证实了当裂纹垂直极化轴时,压电材料的断裂应力随着外加正电场的增加而减小,随着外加负电场的增加而增加．当裂纹平行于极化轴时,与极化轴平行的外加电场对断裂应力无影响     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

热压电介质中的渗透型周期裂纹问题

利用Stroh公式,研究了含共线周期裂纹热的压电介质的广义二维问题。该工作有两个特征：一是裂纹被建模为具有渗透表面的缝隙,并假设为跨越上下表面时,电场的切向分量和电位移的法向分量是连续的;另一个特征是,机－电载荷和热载荷被假设作用在无限远处,而不是在裂纹表面。基于这两个假设,我们获得了有关场强因子,以及裂纹内电场的相当简洁的表达式。结果表明：①在裂纹内电场是线性变化的,②电位移的奇异性总是取决于应力的奇异性.③所有场的奇异性与所加的电载荷无关。     

Surface cracking of a piezoelectric strip bonded to an elastic substrate (Mode I crack problem)

Summary A piezoelectric material layer bonded to an elastic substrate is investigated. The piezoelectric layer contains an edge crack that is perpendicular to the surface of medium. The poling axis of the piezoelectric layer is parallel to its edge. The elastic layer can be an ideal insulator or an ideal conductor. The Fourier transform technique is used to reduce the problem to a solution of singular integral equations. Both impermeable crack and permeable crack assumptions are considered. Stress and electric displacement intensity factors are investigated for different dimensions of the medium. A double-edge cracked symmetric piezoelectric laminate under symmetric electro-mechanical load is also investigated. BLW would like to thank the National Science Foundation of China (#10102004) and the City University of Hong Kong for the support of this work (DAG #7100219). YGS also thanks the Multidiscipline Scientific Research Foundation Project (HIT. MD 2001. 39) of the Harbin Institute of Technology and the SRF for ROCS, SEM.     

Out-of-plane interface cracks in dissimilar piezoelectric materials

Summary This paper investigates the problem of an anti-plane interfacial crack between two dissimilar piezoelectric material layers. A single crack is first considered. The effect of interaction of two collinear cracks in the medium on the field intensity factors is investigated. The solutions of several particular cases, including an infinite piezoelectric bi-material and a piezoelectric material bonded to an elastic medium, are given. The bi-material constants governing the behavior of the crack tip fields are identified. By considering the crack as a notch of finite thickness, it is shown that the thickness of the notch has a pronounced influence on the crack tip field. The results for the assumption of a permeable crack represent the limit case where the notch thickness is reduced to zero.BLW would like to thank the National Science Foundation of China (#10102004) and the City University of Hong Kong (DAG #7100219) for the support of this work. YGS also thanks the Multidiscipline Scientific Research Foundation Project (HIT. MD 2001. 39) of the Harbin Institute of Technology and the SRF for ROCS, SEM.accepted for publication 3 April 2003     

Why traction-free? Piezoelectric crack and Coulombic traction

The non-zero traction condition is introduced in piezoelectric crack problems with the unknown Coulombic traction acting on the crack surfaces. An analytical solution under this condition is obtained by means of the generalized Stroh formalism and by accounting for the permittivity of medium inside the crack gap. As the crack in such materials can be thought of as a low-capacitance medium carrying a potential drop, the Coulombic traction always pulls the two opposite surfaces of the crack together. It is proved that under relatively larger mechanical loading and relatively smaller electrical field, the Coulombic traction may be negligible and the previous investigations under the traction-free crack condition may be accepted in a tolerant way, otherwise the Coulombic traction may lead to some erroneous results with over 10% relative errors. It is also shown that, unlike the traction-free crack condition, the applied electric field does change the Mode I stress intensity factor (SIF) for a central crack in an infinite plane piezoelectric material, and in this way may significantly influence piezoelectric fracture. It is also concluded that the variable tendencies of the normalized SIF and the ERR against the applied electric field depend on the mechanical loading levels. This load-dependence feature may lead to a transformation of the normalized SIF and the ERR from an even functional dependence to an odd functional dependence on the applied electric field.     

PZT-4紧凑拉伸试样的断裂分析

基于线性压电材料的复势理论,通过解析分析,导出了一种分析有限压电板裂纹问题的解析数值方法. 首先,计算了含中心裂纹有限板的断裂参数,与Woo和Wang的解析数值法(Int J Fract, 1993, 62: 203$\sim$218)相比较,表明该方法具有很高的精度和很好的计算效率. 随后,采用该方法和有限元法计算了PZT-4紧凑拉伸试样在绝缘裂纹面边界条件下断裂时的断裂参数,发现各断裂参数的临界值分散性很大,不能作为压电材料的单参数断裂准则. 进而,针对试样真实的裂隙形状,采用有限元法计算了裂隙尖端的应力、电位移场,比较了裂隙内介质的介电性能对裂隙尖端场的影响,计算了带微裂纹的真实裂隙模型的断裂参数并进行了理论分析.      

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.     

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.